Rs Aggarwal 2021 2022 Solutions for Class 10 MATHS Chapter 3

Question 1:

Answer:

On a graph paper, draw a horizontal line X'OX and a vertical line YOY' representing the x-axis and y-axis, respectively.
Graph of 2x + 3y = 2

2x + 3y = 2
⇒ 3y = (2 − 2x)
⇒ 3y = 2(1 − x)
⇒ y = $\frac{2 (1 - x)}{3}$ ...(i)
Putting x = 1, we get y = 0
Putting x = − 2, we get y = 2
Putting x = 4, we get y = −2
Thus, we have the following table for the equation 2x + 3y = 2.

x	1	−2	4
y	0	2	−2

Now, plot the points A(1, 0), B(− 2 , 2) and C(4, − 2) on the graph paper.
Join AB and AC to get the graph line BC. Extend it on both ways.
Thus, the line BC is the graph of 2x + 3y = 2.

Graph of x − 2y = 8
x − 2y = 8
⇒ 2y = ( x − 8)
⇒

y = \frac{x - 8}{2}

...(ii)
Putting x = 2, we get y = −3
Putting x = 4, we get y = −2
Putting x = 0, we get y = −4
Thus, we have the following table for the equation x - 2y = 8.

x	2	4	0
y	− 3	− 2	− 4

Now, plot the points P(0, − 4) and Q(2, − 3). The point C(4, −2) has already been plotted. Join PQ and QC and extend it on both ways.
Thus, line PC is the graph of x − 2y = 8.

The two graph lines intersect at C(4, −2).
∴ x = 4 and y = −2 are the solutions of the given system of equations.

Question 2:

On a graph paper, draw a horizontal line X'OX and a vertical line YOY' representing the x-axis and y-axis, respectively.
Graph of 2x + 3y = 2

2x + 3y = 2
⇒ 3y = (2 − 2x)
⇒ 3y = 2(1 − x)
⇒ y = $\frac{2 (1 - x)}{3}$ ...(i)
Putting x = 1, we get y = 0
Putting x = − 2, we get y = 2
Putting x = 4, we get y = −2
Thus, we have the following table for the equation 2x + 3y = 2.

x	1	−2	4
y	0	2	−2

Now, plot the points A(1, 0), B(− 2 , 2) and C(4, − 2) on the graph paper.
Join AB and AC to get the graph line BC. Extend it on both ways.
Thus, the line BC is the graph of 2x + 3y = 2.

Graph of x − 2y = 8
x − 2y = 8
⇒ 2y = ( x − 8)
⇒

y = \frac{x - 8}{2}

...(ii)
Putting x = 2, we get y = −3
Putting x = 4, we get y = −2
Putting x = 0, we get y = −4
Thus, we have the following table for the equation x - 2y = 8.

x	2	4	0
y	− 3	− 2	− 4

Now, plot the points P(0, − 4) and Q(2, − 3). The point C(4, −2) has already been plotted. Join PQ and QC and extend it on both ways.
Thus, line PC is the graph of x − 2y = 8.

The two graph lines intersect at C(4, −2).
∴ x = 4 and y = −2 are the solutions of the given system of equations.

Answer:

$Given : 3 x + 2 y = 12 . . . (1) x - y + 1 = 0 . . . (2) x - y + 1 = 0 \Rightarrow y = x + 1 . . . (3) Substituting the value of y in (1), we get 3 x + 2 (x + 1) = 12 \Rightarrow 3 x + 2 x + 2 = 12 \Rightarrow 5 x = 12 - 2 \Rightarrow 5 x = 10 \Rightarrow x = 2 . . . (4) Substituting the value of x in (3), we get y = 2 + 1 = 3 Hence, x = 2 and y = 3 .$

Question 3:

$Given : 3 x + 2 y = 12 . . . (1) x - y + 1 = 0 . . . (2) x - y + 1 = 0 \Rightarrow y = x + 1 . . . (3) Substituting the value of y in (1), we get 3 x + 2 (x + 1) = 12 \Rightarrow 3 x + 2 x + 2 = 12 \Rightarrow 5 x = 12 - 2 \Rightarrow 5 x = 10 \Rightarrow x = 2 . . . (4) Substituting the value of x in (3), we get y = 2 + 1 = 3 Hence, x = 2 and y = 3 .$

Answer:

On a graph paper, draw a horizontal line X'OX and a vertical line YOY' as the x-axis and y-axis, respectively.
Graph of 2x + 3y = 8

2x + 3y = 8
⇒ 3y = (8 − 2x)
∴ $y = \frac{8 - 2 x}{3}$ ...........(i)
Putting x = 1, we get y = 2.
Putting x = −5, we get y = 6.
Putting x = 7, we get y = −2.
Thus, we have the following table for the equation 2x + 3y = 8.

x	1	−5	7
y	2	6	−2

Now, plot the points A(1, 2), B(5, −6) and C(7, −2) on the graph paper.
Join AB and AC to get the graph line BC. Extend it on both ways.
Thus, BC is the graph of 2x + 3y = 8.

Graph of x − 2y + 3 = 0
x − 2y + 3 = 0
⇒ 2y = (x + 3)
∴

y = \frac{x + 3}{2}

..........(ii)
Putting x = 1, we get y = 2.
Putting x = 3, we get y = 3.
Putting x = −3, we get y = 0.
Thus, we have the following table for the equation x − 2y + 3 = 0.

x	1	3	−3
y	2	3	0

Now, plot the points P(3, 3) and Q(−3, 0). The point A(1, 2) has already been plotted. Join AP and QA and extend it on both ways.
Thus, PQ is the graph of x − 2y + 3 = 0.

The two graph lines intersect at A(1, 2).
∴ x = 1 and y = 2

Question 4:

On a graph paper, draw a horizontal line X'OX and a vertical line YOY' as the x-axis and y-axis, respectively.
Graph of 2x + 3y = 8

2x + 3y = 8
⇒ 3y = (8 − 2x)
∴ $y = \frac{8 - 2 x}{3}$ ...........(i)
Putting x = 1, we get y = 2.
Putting x = −5, we get y = 6.
Putting x = 7, we get y = −2.
Thus, we have the following table for the equation 2x + 3y = 8.

x	1	−5	7
y	2	6	−2

Now, plot the points A(1, 2), B(5, −6) and C(7, −2) on the graph paper.
Join AB and AC to get the graph line BC. Extend it on both ways.
Thus, BC is the graph of 2x + 3y = 8.

Graph of x − 2y + 3 = 0
x − 2y + 3 = 0
⇒ 2y = (x + 3)
∴

y = \frac{x + 3}{2}

..........(ii)
Putting x = 1, we get y = 2.
Putting x = 3, we get y = 3.
Putting x = −3, we get y = 0.
Thus, we have the following table for the equation x − 2y + 3 = 0.

x	1	3	−3
y	2	3	0

Now, plot the points P(3, 3) and Q(−3, 0). The point A(1, 2) has already been plotted. Join AP and QA and extend it on both ways.
Thus, PQ is the graph of x − 2y + 3 = 0.

The two graph lines intersect at A(1, 2).
∴ x = 1 and y = 2

Answer:

On a graph paper, draw a horizontal line X'OX and a vertical line YOY' as the x-axis and y-axis, respectively.
Graph of 2x − 5y + 4 = 0

2x − 5y + 4 = 0
⇒ 5y = (2x + 4)
∴ $y = \frac{2 x + 4}{5}$ ...........(i)
Putting x = −2, we get y = 0.
Putting x = 3, we get y = 2.
Putting x = 8, we get y = 4.
Thus, we have the following table for the equation 2x − 5y + 4 = 0.

x	−2	3	8
y	0	2	4

Now, plot the points A(−2 , 0), B(3, 2) and C(8, 4) on the graph paper.
Join AB and BC to get the graph line AC. Extend it on both ways.
Thus, AC is the graph of 2x − 5y + 4 = 0.

Graph of 2x + y − 8 = 0
2x + y − 8 = 0
⇒ y = (8 − 2x) ..........(ii)
Putting x = 1, we get y = 6.
Putting x = 3, we get y = 2.
Putting x = 2, we get y = 4.
Thus, we have the following table for the equation 2x + y − 8 = 0.

x	1	3	2
y	6	2	4

Now, plot the points P(1, 6) and Q(2, 4). The point B(3, 2) has already been plotted. Join PQ and QB and extend it on both ways.
Thus, PB is the graph of 2x + y − 8 = 0.

The two graph lines intersect at B(3, 2).
∴ x = 3 and y = 2

Question 5:

On a graph paper, draw a horizontal line X'OX and a vertical line YOY' as the x-axis and y-axis, respectively.
Graph of 2x − 5y + 4 = 0

2x − 5y + 4 = 0
⇒ 5y = (2x + 4)
∴ $y = \frac{2 x + 4}{5}$ ...........(i)
Putting x = −2, we get y = 0.
Putting x = 3, we get y = 2.
Putting x = 8, we get y = 4.
Thus, we have the following table for the equation 2x − 5y + 4 = 0.

x	−2	3	8
y	0	2	4

Now, plot the points A(−2 , 0), B(3, 2) and C(8, 4) on the graph paper.
Join AB and BC to get the graph line AC. Extend it on both ways.
Thus, AC is the graph of 2x − 5y + 4 = 0.

Graph of 2x + y − 8 = 0
2x + y − 8 = 0
⇒ y = (8 − 2x) ..........(ii)
Putting x = 1, we get y = 6.
Putting x = 3, we get y = 2.
Putting x = 2, we get y = 4.
Thus, we have the following table for the equation 2x + y − 8 = 0.

x	1	3	2
y	6	2	4

Now, plot the points P(1, 6) and Q(2, 4). The point B(3, 2) has already been plotted. Join PQ and QB and extend it on both ways.
Thus, PB is the graph of 2x + y − 8 = 0.

The two graph lines intersect at B(3, 2).
∴ x = 3 and y = 2

Answer:

The given equations are:
$3 x + 2 y = 12 . . . . . (i) 5 x - 2 y = 4 . . . . . (i i)$
From (i), write y in terms of x
$y = \frac{12 - 3 x}{2} . . . . . (i i i)$
Now, substitute different values of x in (iii) to get different values of y
For x = 0, $y = \frac{12 - 3 x}{2} = \frac{12 - 0}{2} = 6$
For x = 2, $y = \frac{12 - 3 x}{2} = \frac{12 - 6}{2} = 3$
For x = 4, $y = \frac{12 - 3 x}{2} = \frac{12 - 12}{2} = 0$
Thus, the table for the first equation (3x + 2y = 12) is

x	0	2	4
y	6	3	0

Now, plot the points A(0, 6), B(2, 3) and C(4, 0) on a graph paper and join
A, B and C to get the graph of 3x + 2y = 12.
From (ii), write y in terms of x

y = \frac{5 x - 4}{2} . . . . . (i v)

Now, substitute different values of x in (iv) to get different values of y
For x = 0,

y = \frac{5 x - 4}{2} = \frac{0 - 4}{2} = - 2

For x = 2,

y = \frac{5 x - 4}{2} = \frac{10 - 4}{2} = 3

For x = 4,

y = \frac{5 x - 4}{2} = \frac{20 - 4}{2} = 8

Thus, the table for the first equation (5x − 2y = 4) is

x	0	2	4
y	−2	3	8

Now, plot the points D(0, −2), E(2, 3) and F(4, 8) on the same graph paper and join
D, E and F to get the graph of 5x − 2y = 4.

From the graph it is clear that, the given lines intersect at (2, 3).
Hence, the solution of the given system of equations is (2, 3).

Question 6:

The given equations are:
$3 x + 2 y = 12 . . . . . (i) 5 x - 2 y = 4 . . . . . (i i)$
From (i), write y in terms of x
$y = \frac{12 - 3 x}{2} . . . . . (i i i)$
Now, substitute different values of x in (iii) to get different values of y
For x = 0, $y = \frac{12 - 3 x}{2} = \frac{12 - 0}{2} = 6$
For x = 2, $y = \frac{12 - 3 x}{2} = \frac{12 - 6}{2} = 3$
For x = 4, $y = \frac{12 - 3 x}{2} = \frac{12 - 12}{2} = 0$
Thus, the table for the first equation (3x + 2y = 12) is

x	0	2	4
y	6	3	0

Now, plot the points A(0, 6), B(2, 3) and C(4, 0) on a graph paper and join
A, B and C to get the graph of 3x + 2y = 12.
From (ii), write y in terms of x

y = \frac{5 x - 4}{2} . . . . . (i v)

Now, substitute different values of x in (iv) to get different values of y
For x = 0,

y = \frac{5 x - 4}{2} = \frac{0 - 4}{2} = - 2

For x = 2,

y = \frac{5 x - 4}{2} = \frac{10 - 4}{2} = 3

For x = 4,

y = \frac{5 x - 4}{2} = \frac{20 - 4}{2} = 8

Thus, the table for the first equation (5x − 2y = 4) is

x	0	2	4
y	−2	3	8

Now, plot the points D(0, −2), E(2, 3) and F(4, 8) on the same graph paper and join
D, E and F to get the graph of 5x − 2y = 4.

From the graph it is clear that, the given lines intersect at (2, 3).
Hence, the solution of the given system of equations is (2, 3).

Answer:

n a graph paper, draw a horizontal line X'OX and a vertical line YOY' as the x - axis and y - axis respectively.
Graph of 3x + y + 1 = 0

3x + y + 1 = 0
⇒ y = (−3x − 1) ...........(i)
Putting x = 0, we get y = −1.
Putting x = −1, we get y = 2.
Putting x = 1, we get y = −4.
Thus, we have the following table for the equation 3x + y + 1 = 0.

x	0	−1	1
y	−1	2	−4

Now, plot the points A(0, −1), B(−1, 2) and C(1, −4) on the graph paper.
Join AB and AC to get the graph line BC. Extend it on both ways.
Thus, BC is the graph of 3x + y + 1 = 0.

Graph of 2x − 3y + 8 = 0
2x − 3y + 8 = 0
⇒ 3y = (2x + 8)
∴

y = \frac{2 x + 8}{3}

Putting x = −1, we get y = 2.
Putting x = 2, we get y = 4.
Putting x = −4, we get y = 0.
Thus, we have the following table for the equation 2x − 3y + 8 = 0.

x	−1	2	−4
y	2	4	0

Now, plot the points P(2, 4), Q(−4, 0). The point B(−1 , 2) has already been plotted. Join PB and BQ and extend it on both ways.
Thus, PQ is the graph of 2x − 3y + 8 = 0.

The two graph lines intersect at B(−1, 2).
∴ x = −1 and y = 2

Question 7:

n a graph paper, draw a horizontal line X'OX and a vertical line YOY' as the x - axis and y - axis respectively.
Graph of 3x + y + 1 = 0

3x + y + 1 = 0
⇒ y = (−3x − 1) ...........(i)
Putting x = 0, we get y = −1.
Putting x = −1, we get y = 2.
Putting x = 1, we get y = −4.
Thus, we have the following table for the equation 3x + y + 1 = 0.

x	0	−1	1
y	−1	2	−4

Now, plot the points A(0, −1), B(−1, 2) and C(1, −4) on the graph paper.
Join AB and AC to get the graph line BC. Extend it on both ways.
Thus, BC is the graph of 3x + y + 1 = 0.

Graph of 2x − 3y + 8 = 0
2x − 3y + 8 = 0
⇒ 3y = (2x + 8)
∴

y = \frac{2 x + 8}{3}

Putting x = −1, we get y = 2.
Putting x = 2, we get y = 4.
Putting x = −4, we get y = 0.
Thus, we have the following table for the equation 2x − 3y + 8 = 0.

x	−1	2	−4
y	2	4	0

Now, plot the points P(2, 4), Q(−4, 0). The point B(−1 , 2) has already been plotted. Join PB and BQ and extend it on both ways.
Thus, PQ is the graph of 2x − 3y + 8 = 0.

The two graph lines intersect at B(−1, 2).
∴ x = −1 and y = 2

Answer:

From the first equation, write y in terms of x
$y = - (\frac{5 + 2 x}{3}) . . . . . (i)$
Substitute different values of x in (i) to get different values of y
$For x = - 1, y = - \frac{5 - 2}{3} = - 1 For x = 2, y = - \frac{5 + 4}{3} = - 3 For x = 5, y = - \frac{5 + 10}{3} = - 5$
Thus, the table for the first equation ( 2x + 3y + 5 = 0 ) is

x	−1	2	5
y	−1	−3	−5

Now, plot the points A(−1,−1), B(2,−3) and C(5,−5) on a graph paper and join
them to get the graph of 2x + 3y + 5 = 0.
From the second equation, write y in terms of x

y = \frac{3 x - 12}{2} . . . . . (i i)

Now, substitute different values of x in (ii) to get different values of y

For x = 0, y = \frac{0 - 12}{2} = - 6 For x = 2, y = \frac{6 - 12}{2} = - 3 For x = 4, y = \frac{12 - 12}{2} = 0

So, the table for the second equation ( 3x − 2y − 12 = 0 ) is

x	0	2	4
y	−6	−3	0

Now, plot the points D(0,−6), E(2,−3) and F(4,0) on the same graph paper and join
D, E and F to get the graph of 3x − 2y − 12 = 0.

From the graph it is clear that, the given lines intersect at (2,−3).
Hence, the solution of the given system of equation is (2,−3).

Question 8:

From the first equation, write y in terms of x
$y = - (\frac{5 + 2 x}{3}) . . . . . (i)$
Substitute different values of x in (i) to get different values of y
$For x = - 1, y = - \frac{5 - 2}{3} = - 1 For x = 2, y = - \frac{5 + 4}{3} = - 3 For x = 5, y = - \frac{5 + 10}{3} = - 5$
Thus, the table for the first equation ( 2x + 3y + 5 = 0 ) is

x	−1	2	5
y	−1	−3	−5

Now, plot the points A(−1,−1), B(2,−3) and C(5,−5) on a graph paper and join
them to get the graph of 2x + 3y + 5 = 0.
From the second equation, write y in terms of x

y = \frac{3 x - 12}{2} . . . . . (i i)

Now, substitute different values of x in (ii) to get different values of y

For x = 0, y = \frac{0 - 12}{2} = - 6 For x = 2, y = \frac{6 - 12}{2} = - 3 For x = 4, y = \frac{12 - 12}{2} = 0

So, the table for the second equation ( 3x − 2y − 12 = 0 ) is

x	0	2	4
y	−6	−3	0

Now, plot the points D(0,−6), E(2,−3) and F(4,0) on the same graph paper and join
D, E and F to get the graph of 3x − 2y − 12 = 0.

From the graph it is clear that, the given lines intersect at (2,−3).
Hence, the solution of the given system of equation is (2,−3).

Answer:

From the first equation, write y in terms of x
$y = \frac{2 x + 13}{3} . . . . . (i)$
Substitute different values of x in (i) to get different values of y
$For x = - 5, y = \frac{- 10 + 13}{3} = 1 For x = 1, y = \frac{2 + 13}{3} = 5 For x = 4, y = \frac{8 + 13}{3} = 7$
Thus, the table for the first equation ( 2x − 3y + 13 = 0 ) is

x	−5	1	4
y	1	5	7

Now, plot the points A(−5,1), B(1,5) and C(4,7) on a graph paper and join
A, B and C to get the graph of 2x − 3y + 13 = 0.
From the second equation, write y in terms of x

y = \frac{3 x + 12}{2} . . . . . (i i)

Now, substitute different values of x in (ii) to get different values of y

For x = - 4, y = \frac{- 12 + 12}{2} = 0 For x = - 2, y = \frac{- 6 + 12}{2} = 3 For x = 0, y = \frac{0 + 12}{2} = 6

So, the table for the second equation ( 3x − 2y + 12 = 0 ) is

x	−4	−2	0
y	0	3	6

Now, plot the points D(−4,0), E(−2,3) and F(0,6) on the same graph paper and join
D, E and F to get the graph of 3x − 2y + 12 = 0.

From the graph it is clear that, the given lines intersect at (−2,3).
Hence, the solution of the given system of equation is (−2,3).

Question 9:

From the first equation, write y in terms of x
$y = \frac{2 x + 13}{3} . . . . . (i)$
Substitute different values of x in (i) to get different values of y
$For x = - 5, y = \frac{- 10 + 13}{3} = 1 For x = 1, y = \frac{2 + 13}{3} = 5 For x = 4, y = \frac{8 + 13}{3} = 7$
Thus, the table for the first equation ( 2x − 3y + 13 = 0 ) is

x	−5	1	4
y	1	5	7

Now, plot the points A(−5,1), B(1,5) and C(4,7) on a graph paper and join
A, B and C to get the graph of 2x − 3y + 13 = 0.
From the second equation, write y in terms of x

y = \frac{3 x + 12}{2} . . . . . (i i)

Now, substitute different values of x in (ii) to get different values of y

For x = - 4, y = \frac{- 12 + 12}{2} = 0 For x = - 2, y = \frac{- 6 + 12}{2} = 3 For x = 0, y = \frac{0 + 12}{2} = 6

So, the table for the second equation ( 3x − 2y + 12 = 0 ) is

x	−4	−2	0
y	0	3	6

Now, plot the points D(−4,0), E(−2,3) and F(0,6) on the same graph paper and join
D, E and F to get the graph of 3x − 2y + 12 = 0.

From the graph it is clear that, the given lines intersect at (−2,3).
Hence, the solution of the given system of equation is (−2,3).

Answer:

On a graph paper, draw a horizontal line X'OX and a vertical line YOY' as the x-axis and y-axis, respectively.
Graph of 2x + 3y = 4

2x + 3y = 4
⇒ 3y = (4 − 2x)
∴ $y = \frac{4 - 2 x}{3}$ ...(i)
Putting x = −1, we get y = 2
Putting x = 2, we get y = 0
Putting x = 5, we get y = −2
Thus, we have the following table for the equation 2x + 3y = 4.

x	−1	2	5
y	2	0	−2

Now, plot the points A(−1 , 2), B(2 , 0) and C(5, −2) on the graph paper.
Join AB and BC to get the graph line AC. Extend it on both ways.
Thus, AC is the graph of 2x + 3y = 4.

                                                 Graph of 3x − y = −5
3x − y = −5
⇒ y = (3x + 5)              ...(ii)
Putting x = −1, we get y = 2
Putting x = 0, we get y = 5
Putting x = −2, we get y = −1
Thus, we have the following table for the equation 3x − y = − 5 = 0.

x	−1	0	−2
y	2	5	−1

Now, plot the points P(0, 5), Q(−2 , −1). The point A(−1 , 2) has already been plotted. Join PA and QA and extend it on both ways.
Thus, PQ is the graph of 3x − y = −5.

The two graph lines intersect at A(−1 , 2).
∴ x = −1 and y = 2 are the solutions of the given system of equations.

Question 10:

On a graph paper, draw a horizontal line X'OX and a vertical line YOY' as the x-axis and y-axis, respectively.
Graph of 2x + 3y = 4

2x + 3y = 4
⇒ 3y = (4 − 2x)
∴ $y = \frac{4 - 2 x}{3}$ ...(i)
Putting x = −1, we get y = 2
Putting x = 2, we get y = 0
Putting x = 5, we get y = −2
Thus, we have the following table for the equation 2x + 3y = 4.

x	−1	2	5
y	2	0	−2

Now, plot the points A(−1 , 2), B(2 , 0) and C(5, −2) on the graph paper.
Join AB and BC to get the graph line AC. Extend it on both ways.
Thus, AC is the graph of 2x + 3y = 4.

                                                 Graph of 3x − y = −5
3x − y = −5
⇒ y = (3x + 5)              ...(ii)
Putting x = −1, we get y = 2
Putting x = 0, we get y = 5
Putting x = −2, we get y = −1
Thus, we have the following table for the equation 3x − y = − 5 = 0.

x	−1	0	−2
y	2	5	−1

Now, plot the points P(0, 5), Q(−2 , −1). The point A(−1 , 2) has already been plotted. Join PA and QA and extend it on both ways.
Thus, PQ is the graph of 3x − y = −5.

The two graph lines intersect at A(−1 , 2).
∴ x = −1 and y = 2 are the solutions of the given system of equations.

Answer:

On a graph paper, draw a horizontal line X'OX and a vertical line YOY' as the x-axis and y-axis, respectively.
Graph of x + 2y + 2 = 0

x + 2y + 2 = 0
⇒ 2y = (−2 − x)
∴ $y = \frac{- 2 - x}{2}$ ...............(i)
Putting x = −2, we get y = 0.
Putting x = 0, we get y = −1.
Putting x = 2, we get y = −2.
Thus, we have the following table for the equation x + 2y + 2 = 0.

x	−2	0	2
y	0	−1	−2

Now, plot the points A(−2, 0) , B(0 , −1) and C(2, −2) on the graph paper.
Join AB and BC to get the graph line AC. Extend it on both ways.
Thus, AC is the graph of x + 2y + 2 = 0.

Graph of 3x + 2y − 2 = 0
3x + 2y − 2 = 0
⇒ 2y = (2 − 3x)
∴

y = \frac{2 - 3 x}{2}

...............(ii)
Putting x = 0, we get y = 1.
Putting x = 2, we get y = −2.
Putting x = 4, we get y = −5.
Thus, we have the following table for the equation 3x + 2y − 2 = 0.

x	0	2	4
y	1	−2	−5

Now, plot the points P(0, 1) and Q(4, −5). The point C(2, −2) has already been plotted. Join PC and QC and extend it on both ways.
Thus, PQ is the graph of 3x + 2y − 2 = 0.

The two graph lines intersect at A(2, −2).
∴ x = 2 and y = −2

Question 11:

On a graph paper, draw a horizontal line X'OX and a vertical line YOY' as the x-axis and y-axis, respectively.
Graph of x + 2y + 2 = 0

x + 2y + 2 = 0
⇒ 2y = (−2 − x)
∴ $y = \frac{- 2 - x}{2}$ ...............(i)
Putting x = −2, we get y = 0.
Putting x = 0, we get y = −1.
Putting x = 2, we get y = −2.
Thus, we have the following table for the equation x + 2y + 2 = 0.

x	−2	0	2
y	0	−1	−2

Now, plot the points A(−2, 0) , B(0 , −1) and C(2, −2) on the graph paper.
Join AB and BC to get the graph line AC. Extend it on both ways.
Thus, AC is the graph of x + 2y + 2 = 0.

Graph of 3x + 2y − 2 = 0
3x + 2y − 2 = 0
⇒ 2y = (2 − 3x)
∴

y = \frac{2 - 3 x}{2}

...............(ii)
Putting x = 0, we get y = 1.
Putting x = 2, we get y = −2.
Putting x = 4, we get y = −5.
Thus, we have the following table for the equation 3x + 2y − 2 = 0.

x	0	2	4
y	1	−2	−5

Now, plot the points P(0, 1) and Q(4, −5). The point C(2, −2) has already been plotted. Join PC and QC and extend it on both ways.
Thus, PQ is the graph of 3x + 2y − 2 = 0.

The two graph lines intersect at A(2, −2).
∴ x = 2 and y = −2

Answer:

From the first equation, write y in terms of x
$y = x + 3 . . . . . (i)$
Substitute different values of x in (i) to get different values of y
$For x = - 3, y = - 3 + 3 = 0 For x = - 1, y = - 1 + 3 = 2 For x = 1, y = 1 + 3 = 4$
Thus, the table for the first equation (x − y + 3 = 0) is

x	−3	−1	1
y	0	2	4

Now, plot the points A(−3,0), B(−1,2) and C(1,4) on a graph paper and join
A, B and C to get the graph of x − y + 3 = 0.
From the second equation, write y in terms of x

y = \frac{4 - 2 x}{3} . . . . . (i i)

Now, substitute different values of x in (ii) to get different values of y

For x = - 4, y = \frac{4 + 8}{3} = 4 For x = - 1, y = \frac{4 + 2}{3} = 2 For x = 2, y = \frac{4 - 4}{3} = 0

So, the table for the second equation ( 2x + 3y − 4 = 0 ) is

x	−4	−1	2
y	4	2	0

Now, plot the points D(−4,4), E(−1,2) and F(2,0) on the same graph paper and join
D, E and F to get the graph of 2x − 3y − 4 = 0.

From the graph it is clear that, the given lines intersect at (−1,2).
So, the solution of the given system of equation is (−1,2).
The vertices of the triangle formed by the given lines and the x-axis are (−3,0), (−1,2) and (2,0).
Now, draw a perpendicular from the intersection point E on the x-axis. So,

Area (∆ E A F) = \frac{1}{2} \times A F \times E M = \frac{1}{2} \times 5 \times 2 = 5 sq . units

Hence, the vertices of the triangle formed by the given lines and the x-axis are (−3,0), (−1,2)
and (2,0) and its area is 5 sq. units.

Question 12:

From the first equation, write y in terms of x
$y = x + 3 . . . . . (i)$
Substitute different values of x in (i) to get different values of y
$For x = - 3, y = - 3 + 3 = 0 For x = - 1, y = - 1 + 3 = 2 For x = 1, y = 1 + 3 = 4$
Thus, the table for the first equation (x − y + 3 = 0) is

x	−3	−1	1
y	0	2	4

Now, plot the points A(−3,0), B(−1,2) and C(1,4) on a graph paper and join
A, B and C to get the graph of x − y + 3 = 0.
From the second equation, write y in terms of x

y = \frac{4 - 2 x}{3} . . . . . (i i)

Now, substitute different values of x in (ii) to get different values of y

For x = - 4, y = \frac{4 + 8}{3} = 4 For x = - 1, y = \frac{4 + 2}{3} = 2 For x = 2, y = \frac{4 - 4}{3} = 0

So, the table for the second equation ( 2x + 3y − 4 = 0 ) is

x	−4	−1	2
y	4	2	0

Now, plot the points D(−4,4), E(−1,2) and F(2,0) on the same graph paper and join
D, E and F to get the graph of 2x − 3y − 4 = 0.

From the graph it is clear that, the given lines intersect at (−1,2).
So, the solution of the given system of equation is (−1,2).
The vertices of the triangle formed by the given lines and the x-axis are (−3,0), (−1,2) and (2,0).
Now, draw a perpendicular from the intersection point E on the x-axis. So,

Area (∆ E A F) = \frac{1}{2} \times A F \times E M = \frac{1}{2} \times 5 \times 2 = 5 sq . units

Hence, the vertices of the triangle formed by the given lines and the x-axis are (−3,0), (−1,2)
and (2,0) and its area is 5 sq. units.

Answer:

From the first equation, write y in terms of x
$y = \frac{2 x + 4}{3} . . . . . (i)$
Substitute different values of x in (i) to get different values of y
$For x = - 2, y = \frac{- 4 + 4}{3} = 0 For x = 1, y = \frac{2 + 4}{3} = 2 For x = 4, y = \frac{8 + 4}{3} = 4$
Thus, the table for the first equation (2x − 3y + 4 = 0) is

x	−2	1	4
y	0	2	4

Now, plot the points A(−2,0), B(1,2) and C(4,4) on a graph paper and join
A, B and C to get the graph of 2x − 3y + 4 = 0.
From the second equation, write y in terms of x

y = \frac{5 - x}{2} . . . . . (i i)

Now, substitute different values of x in (ii) to get different values of y

For x = - 3, y = \frac{5 + 3}{2} = 4 For x = 1, y = \frac{5 - 1}{2} = 2 For x = 5, y = \frac{5 - 5}{2} = 0

So, the table for the second equation ( x + 2y − 5 = 0 ) is

x	−3	1	5
y	4	2	0

Now, plot the points D(−3,4), B(1,2) and F(5,0) on the same graph paper and join
D, E and F to get the graph of x + 2y − 5 = 0.

From the graph it is clear that, the given lines intersect at (1,2).
So, the solution of the given system of equations is (1,2).
From the graph, the vertices of the triangle formed by the given lines and the x-axis are
(−2,0), (1,2) and (5,0).
Now, draw a perpendicular from the intersection point B on the x-axis. So,

Area (∆ B A F) = \frac{1}{2} \times A F \times B M = \frac{1}{2} \times 7 \times 2 = 7 sq . units

Hence, the vertices of the triangle formed by the given lines and the x-axis are (−2,0), (1,2) and (5,0)
and the area of the triangle is 7 sq. units.

Question 13:

From the first equation, write y in terms of x
$y = \frac{2 x + 4}{3} . . . . . (i)$
Substitute different values of x in (i) to get different values of y
$For x = - 2, y = \frac{- 4 + 4}{3} = 0 For x = 1, y = \frac{2 + 4}{3} = 2 For x = 4, y = \frac{8 + 4}{3} = 4$
Thus, the table for the first equation (2x − 3y + 4 = 0) is

x	−2	1	4
y	0	2	4

Now, plot the points A(−2,0), B(1,2) and C(4,4) on a graph paper and join
A, B and C to get the graph of 2x − 3y + 4 = 0.
From the second equation, write y in terms of x

y = \frac{5 - x}{2} . . . . . (i i)

Now, substitute different values of x in (ii) to get different values of y

For x = - 3, y = \frac{5 + 3}{2} = 4 For x = 1, y = \frac{5 - 1}{2} = 2 For x = 5, y = \frac{5 - 5}{2} = 0

So, the table for the second equation ( x + 2y − 5 = 0 ) is

x	−3	1	5
y	4	2	0

Now, plot the points D(−3,4), B(1,2) and F(5,0) on the same graph paper and join
D, E and F to get the graph of x + 2y − 5 = 0.

From the graph it is clear that, the given lines intersect at (1,2).
So, the solution of the given system of equations is (1,2).
From the graph, the vertices of the triangle formed by the given lines and the x-axis are
(−2,0), (1,2) and (5,0).
Now, draw a perpendicular from the intersection point B on the x-axis. So,

Area (∆ B A F) = \frac{1}{2} \times A F \times B M = \frac{1}{2} \times 7 \times 2 = 7 sq . units

Hence, the vertices of the triangle formed by the given lines and the x-axis are (−2,0), (1,2) and (5,0)
and the area of the triangle is 7 sq. units.

Answer:

On a graph paper, draw a horizontal line X'OX and a vertical line YOY' as the x-axis and y-axis, respectively.
Graph of 4x − 3y + 4 = 0
4x − 3y + 4 = 0
⇒ 3y = (4x + 4)
∴ $y = \frac{4 x + 4}{3}$ ............(i)
Putting x = −1, we get y = 0.
Putting x = 2, we get y = 4.
Putting x = 5, we get y = 8.
Thus, we have the following table for the equation 4x − 3y + 4 = 0

x	−1	2	5
y	0	4	8

Now, plot the points A(−1, 0), B( 2, 4) and C(5, 8) on the graph paper.
Join AB and BC to get the graph line AC. Extend it on both ways.
Thus, AC is the graph of 4x − 3y + 4 = 0.

Graph of 4x + 3y − 20 = 0
4x + 3y − 20 = 0
⇒ 3y = (−4x + 20)
∴

y = \frac{- 4 x + 20}{3}

............(ii)
Putting x = 2, we get y = 4.
Putting x = −1, we get y = 8.
Putting x = 5, we get y = 0.
Thus, we have the following table for the equation 4x + 3y − 20 = 0.

x	2	−1	5
y	4	8	0

Now, plot the points P(1, −8) and Q(5 , 0). The point B(2, 4) has already been plotted. Join PB and QB to get the graph line PQ. Extend it on both ways.
Then, line PQ is the graph of the equation 4x + 3y − 20 = 0.

The two graph lines intersect at B(2, 4).
∴ The solution of the given system of equations is x = 2 and y = 4.
Clearly, the vertices of ΔABQ formed by these two lines and x-axis are Q(5, 0), B(2, 4) and A(−1, 0).
Now, consider ΔABQ.
Here, height = 4 units and base (AQ) = 6 units
∴ Area of ΔABQ =

\frac{1}{2} \times base \times height

sq. units
=

(\frac{1}{2} \times 6 \times 4) = 12

sq. units.

Question 14:

On a graph paper, draw a horizontal line X'OX and a vertical line YOY' as the x-axis and y-axis, respectively.
Graph of 4x − 3y + 4 = 0
4x − 3y + 4 = 0
⇒ 3y = (4x + 4)
∴ $y = \frac{4 x + 4}{3}$ ............(i)
Putting x = −1, we get y = 0.
Putting x = 2, we get y = 4.
Putting x = 5, we get y = 8.
Thus, we have the following table for the equation 4x − 3y + 4 = 0

x	−1	2	5
y	0	4	8

Now, plot the points A(−1, 0), B( 2, 4) and C(5, 8) on the graph paper.
Join AB and BC to get the graph line AC. Extend it on both ways.
Thus, AC is the graph of 4x − 3y + 4 = 0.

Graph of 4x + 3y − 20 = 0
4x + 3y − 20 = 0
⇒ 3y = (−4x + 20)
∴

y = \frac{- 4 x + 20}{3}

............(ii)
Putting x = 2, we get y = 4.
Putting x = −1, we get y = 8.
Putting x = 5, we get y = 0.
Thus, we have the following table for the equation 4x + 3y − 20 = 0.

x	2	−1	5
y	4	8	0

Now, plot the points P(1, −8) and Q(5 , 0). The point B(2, 4) has already been plotted. Join PB and QB to get the graph line PQ. Extend it on both ways.
Then, line PQ is the graph of the equation 4x + 3y − 20 = 0.

The two graph lines intersect at B(2, 4).
∴ The solution of the given system of equations is x = 2 and y = 4.
Clearly, the vertices of ΔABQ formed by these two lines and x-axis are Q(5, 0), B(2, 4) and A(−1, 0).
Now, consider ΔABQ.
Here, height = 4 units and base (AQ) = 6 units
∴ Area of ΔABQ =

\frac{1}{2} \times base \times height

sq. units
=

(\frac{1}{2} \times 6 \times 4) = 12

sq. units.

Answer:

On a graph paper, draw a horizontal line X'OX and a vertical line YOY' as the x-axis and y-axis, respectively.
Graph of x − y + 1 = 0
x − y + 1 = 0
⇒ y = (x + 1) .........(i)
Putting x = −1, we get y = 0.
Putting x = 1, we get y = 2.
Putting x = 2, we get y = 3.
Thus, we have the following table for the equation x − y + 1 = 0.

x	−1	1	2
y	0	2	3

Now, plot the points A(-1, 0), B( 1, 2) and C(2, 3) on the graph paper.
Join AB and BC to get the graph line AC. Extend it on both ways.
Thus, AC is the graph of x − y + 1 = 0.

Graph of 3x + 2y − 12 = 0
3x + 2y − 12 = 0
⇒ 2y = (−3x + 12)
∴

y = \frac{- 3 x + 12}{2}

............(ii)
Putting x = 0, we get y = 6.
Putting x = 2, we get y = 3.
Putting x = 4, we get y = 0.
Thus, we have the following table for the equation 3x + 2y − 12 = 0.

x	0	2	4
y	6	3	0

Now, plot the points P(0, 6) and Q(4 , 0). The point C(2, 3) has already been plotted. Join PC and CQ to get the graph line PQ. Extend it on both ways.
Then, PQ is the graph of the equation 3x + 2y − 12 = 0.

The two graph lines intersect at C(2, 3).
∴ The solution of the given system of equations is x = 2 and y = 3.
Clearly, the vertices of ΔACQ formed by these two lines and the x-axis are Q(4, 0), C(2, 3) and A(−1, 0).
Now, consider ΔACQ.
Here, height = 3 units and base (AQ) = 5 units
Area of ΔACQ =

\frac{1}{2} \times base \times height

sq. units
=

(\frac{1}{2} \times 5 \times 3) = 7.5

sq. units.

Question 15:

On a graph paper, draw a horizontal line X'OX and a vertical line YOY' as the x-axis and y-axis, respectively.
Graph of x − y + 1 = 0
x − y + 1 = 0
⇒ y = (x + 1) .........(i)
Putting x = −1, we get y = 0.
Putting x = 1, we get y = 2.
Putting x = 2, we get y = 3.
Thus, we have the following table for the equation x − y + 1 = 0.

x	−1	1	2
y	0	2	3

Now, plot the points A(-1, 0), B( 1, 2) and C(2, 3) on the graph paper.
Join AB and BC to get the graph line AC. Extend it on both ways.
Thus, AC is the graph of x − y + 1 = 0.

Graph of 3x + 2y − 12 = 0
3x + 2y − 12 = 0
⇒ 2y = (−3x + 12)
∴

y = \frac{- 3 x + 12}{2}

............(ii)
Putting x = 0, we get y = 6.
Putting x = 2, we get y = 3.
Putting x = 4, we get y = 0.
Thus, we have the following table for the equation 3x + 2y − 12 = 0.

x	0	2	4
y	6	3	0

Now, plot the points P(0, 6) and Q(4 , 0). The point C(2, 3) has already been plotted. Join PC and CQ to get the graph line PQ. Extend it on both ways.
Then, PQ is the graph of the equation 3x + 2y − 12 = 0.

The two graph lines intersect at C(2, 3).
∴ The solution of the given system of equations is x = 2 and y = 3.
Clearly, the vertices of ΔACQ formed by these two lines and the x-axis are Q(4, 0), C(2, 3) and A(−1, 0).
Now, consider ΔACQ.
Here, height = 3 units and base (AQ) = 5 units
Area of ΔACQ =

\frac{1}{2} \times base \times height

sq. units
=

(\frac{1}{2} \times 5 \times 3) = 7.5

sq. units.

Answer:

From the first equation, write y in terms of x
$y = \frac{x + 2}{2} . . . . . (i)$
Substitute different values of x in (i) to get different values of y
$For x = - 2, y = \frac{- 2 + 2}{2} = 0 For x = 2, y = \frac{2 + 2}{2} = 2 For x = 4, y = \frac{4 + 2}{2} = 3$
Thus, the table for the first equation (x − 2y + 2 = 0) is

x	−2	2	4
y	0	2	3

Now, plot the points A(−2,0), B(2,2) and C(4,3) on a graph paper and join
A, B and C to get the graph of x − 2y + 2 = 0.
From the second equation, write y in terms of x

y = 6 - 2 x . . . . . (i i)

Now, substitute different values of x in (ii) to get different values of y

For x = 1, y = 6 - 2 = 4 For x = 3, y = 0 For x = 4, y = 6 - 8 = - 2

So, the table for the second equation (2x + y − 6 = 0 ) is

x	1	3	4
y	4	0	−2

Now, plot the points D(1,4), E(3,0) and F(4,−2) on the same graph paper and join
D, E and F to get the graph of 2x + y − 6 = 0.

From the graph it is clear that, the given lines intersect at (2,2).
So, the solution of the given system of equations is (2,2).
From the graph, the vertices of the triangle formed by the given lines and the x-axis are
(−2,0), (2,2) and (3,0).
Now, draw a perpendicular from the intersection point B on the x-axis. So,

Area (∆ B A E) = \frac{1}{2} \times A E \times B M = \frac{1}{2} \times 5 \times 2 = 5 sq . units

Hence, the vertices of the triangle formed by the given lines and the x-axis are
(−2,0), (2,2) and (3,0) and the area of the triangle is 5 sq. units.

Question 16:

From the first equation, write y in terms of x
$y = \frac{x + 2}{2} . . . . . (i)$
Substitute different values of x in (i) to get different values of y
$For x = - 2, y = \frac{- 2 + 2}{2} = 0 For x = 2, y = \frac{2 + 2}{2} = 2 For x = 4, y = \frac{4 + 2}{2} = 3$
Thus, the table for the first equation (x − 2y + 2 = 0) is

x	−2	2	4
y	0	2	3

Now, plot the points A(−2,0), B(2,2) and C(4,3) on a graph paper and join
A, B and C to get the graph of x − 2y + 2 = 0.
From the second equation, write y in terms of x

y = 6 - 2 x . . . . . (i i)

Now, substitute different values of x in (ii) to get different values of y

For x = 1, y = 6 - 2 = 4 For x = 3, y = 0 For x = 4, y = 6 - 8 = - 2

So, the table for the second equation (2x + y − 6 = 0 ) is

x	1	3	4
y	4	0	−2

Now, plot the points D(1,4), E(3,0) and F(4,−2) on the same graph paper and join
D, E and F to get the graph of 2x + y − 6 = 0.

From the graph it is clear that, the given lines intersect at (2,2).
So, the solution of the given system of equations is (2,2).
From the graph, the vertices of the triangle formed by the given lines and the x-axis are
(−2,0), (2,2) and (3,0).
Now, draw a perpendicular from the intersection point B on the x-axis. So,

Area (∆ B A E) = \frac{1}{2} \times A E \times B M = \frac{1}{2} \times 5 \times 2 = 5 sq . units

Hence, the vertices of the triangle formed by the given lines and the x-axis are
(−2,0), (2,2) and (3,0) and the area of the triangle is 5 sq. units.

Answer:

From the first equation, write y in terms of x
$y = \frac{2 x + 6}{3} . . . . . (i)$
Substitute different values of x in (i) to get different values of y
$For x = - 3, y = \frac{- 6 + 6}{3} = 0 For x = 0, y = \frac{0 + 6}{3} = 2 For x = 3, y = \frac{6 + 6}{3} = 4$
Thus, the table for the first equation (2x − 3y + 6 = 0) is

x	−3	0	3
y	0	2	4

Now, plot the points A(−3,0), B(0,2) and C(3,4) on a graph paper and join
A, B and C to get the graph of 2x − 3y + 6 = 0.
From the second equation, write y in terms of x

y = \frac{18 - 2 x}{3} . . . . . (i i)

Now, substitute different values of x in (ii) to get different values of y

For x = 0, y = \frac{18 - 0}{3} = 6 For x = 3, y = \frac{18 - 6}{3} = 4 For x = 9, y = \frac{18 - 18}{3} = 0

So, the table for the second equation (2x + 3y − 18 = 0 ) is

x	0	3	9
y	6	4	0

Now, plot the points D(0,6), E(3,4) and F(9,0) on the same graph paper and join
D, E and F to get the graph of 2x + 3y − 18 = 0.

From the graph it is clear that, the given lines intersect at (3,4).
So, the solution of the given system of equations is (3,4).
From the graph, the vertices of the triangle formed by the given lines and the y-axis are
(0,2), (0,6) and (3,4).
Now, draw a perpendicular from the intersection point E (or C) on the y-axis. So,

Area (∆ E D B) = \frac{1}{2} \times B D \times E M = \frac{1}{2} \times 4 \times 3 = 6 sq . units

Hence, the vertices of the triangle formed by the given lines and the y-axis are
(0,2), (0,6) and (3,4) and its area is 6 sq. units.

Question 17:

From the first equation, write y in terms of x
$y = \frac{2 x + 6}{3} . . . . . (i)$
Substitute different values of x in (i) to get different values of y
$For x = - 3, y = \frac{- 6 + 6}{3} = 0 For x = 0, y = \frac{0 + 6}{3} = 2 For x = 3, y = \frac{6 + 6}{3} = 4$
Thus, the table for the first equation (2x − 3y + 6 = 0) is

x	−3	0	3
y	0	2	4

Now, plot the points A(−3,0), B(0,2) and C(3,4) on a graph paper and join
A, B and C to get the graph of 2x − 3y + 6 = 0.
From the second equation, write y in terms of x

y = \frac{18 - 2 x}{3} . . . . . (i i)

Now, substitute different values of x in (ii) to get different values of y

For x = 0, y = \frac{18 - 0}{3} = 6 For x = 3, y = \frac{18 - 6}{3} = 4 For x = 9, y = \frac{18 - 18}{3} = 0

So, the table for the second equation (2x + 3y − 18 = 0 ) is

x	0	3	9
y	6	4	0

Now, plot the points D(0,6), E(3,4) and F(9,0) on the same graph paper and join
D, E and F to get the graph of 2x + 3y − 18 = 0.

From the graph it is clear that, the given lines intersect at (3,4).
So, the solution of the given system of equations is (3,4).
From the graph, the vertices of the triangle formed by the given lines and the y-axis are
(0,2), (0,6) and (3,4).
Now, draw a perpendicular from the intersection point E (or C) on the y-axis. So,

Area (∆ E D B) = \frac{1}{2} \times B D \times E M = \frac{1}{2} \times 4 \times 3 = 6 sq . units

Hence, the vertices of the triangle formed by the given lines and the y-axis are
(0,2), (0,6) and (3,4) and its area is 6 sq. units.

Answer:

From the first equation, write y in terms of x
$y = 4 x - 4 . . . . . (i)$
Substitute different values of x in (i) to get different values of y
$For x = 0, y = 0 - 4 = - 4 For x = 1, y = 4 - 4 = 0 For x = 2, y = 8 - 4 = 4$
Thus, the table for the first equation (4x − y − 4 = 0) is

x	0	1	2
y	−4	0	4

Now, plot the points A(0,−4), B(1,0) and C(2,4) on a graph paper and join
A, B and C to get the graph of 4x − y − 4 = 0.
From the second equation, write y in terms of x

y = \frac{14 - 3 x}{2} . . . . . (i i)

Now, substitute different values of x in (ii) to get different values of y

For x = 0, y = \frac{14 - 0}{2} = 7 For x = 4, y = \frac{14 - 12}{2} = 1 For x = \frac{14}{3}, y = \frac{14 - 14}{2} = 0

So, the table for the second equation (3x + 2y − 14 = 0 ) is

x	0	4	$\frac{14}{3}$
y	7	1	0

Now, plot the points D(0,7), E(4,1) and

F (\frac{14}{3}, 0)

on the same graph paper and join
D, E and F to get the graph of 3x + 2y − 14 = 0.

From the graph it is clear that, the given lines intersect at (,4).
So, the solution of the given system of equations is (2,4).
From the graph, the vertices of the triangle formed by the given lines and the y-axis are
(0,7), (0, −4) and (2,4).
Now, draw a perpendicular from the intersection point C on the y-axis. So,

Area (∆ D A C) = \frac{1}{2} \times D A \times C M = \frac{1}{2} \times 11 \times 2 = 11 sq . units

Hence, the vertices of the triangle formed by the given lines and the y-axis are
(0,7), (0, −4) and (2,4) and its area is 11 sq. units.

Question 18:

From the first equation, write y in terms of x
$y = 4 x - 4 . . . . . (i)$
Substitute different values of x in (i) to get different values of y
$For x = 0, y = 0 - 4 = - 4 For x = 1, y = 4 - 4 = 0 For x = 2, y = 8 - 4 = 4$
Thus, the table for the first equation (4x − y − 4 = 0) is

x	0	1	2
y	−4	0	4

Now, plot the points A(0,−4), B(1,0) and C(2,4) on a graph paper and join
A, B and C to get the graph of 4x − y − 4 = 0.
From the second equation, write y in terms of x

y = \frac{14 - 3 x}{2} . . . . . (i i)

Now, substitute different values of x in (ii) to get different values of y

For x = 0, y = \frac{14 - 0}{2} = 7 For x = 4, y = \frac{14 - 12}{2} = 1 For x = \frac{14}{3}, y = \frac{14 - 14}{2} = 0

So, the table for the second equation (3x + 2y − 14 = 0 ) is

x	0	4	$\frac{14}{3}$
y	7	1	0

Now, plot the points D(0,7), E(4,1) and

F (\frac{14}{3}, 0)

on the same graph paper and join
D, E and F to get the graph of 3x + 2y − 14 = 0.

From the graph it is clear that, the given lines intersect at (,4).
So, the solution of the given system of equations is (2,4).
From the graph, the vertices of the triangle formed by the given lines and the y-axis are
(0,7), (0, −4) and (2,4).
Now, draw a perpendicular from the intersection point C on the y-axis. So,

Area (∆ D A C) = \frac{1}{2} \times D A \times C M = \frac{1}{2} \times 11 \times 2 = 11 sq . units

Hence, the vertices of the triangle formed by the given lines and the y-axis are
(0,7), (0, −4) and (2,4) and its area is 11 sq. units.

Answer:

From the first equation, write y in terms of x
$y = x - 5 . . . . . (i)$
Substitute different values of x in (i) to get different values of y
$For x = 0, y = 0 - 5 = - 5 For x = 2, y = 2 - 5 = - 3 For x = 5, y = 5 - 5 = 0$
Thus, the table for the first equation (x − y − 5 = 0) is

x	0	2	5
y	−5	−3	0

Now, plot the points A(0,−5), B(2,−3) and C(5,0) on a graph paper and join
A, B and C to get the graph of x − y − 5 = 0.
From the second equation, write y in terms of x

y = \frac{15 - 3 x}{5} . . . . . (i i)

Now, substitute different values of x in (ii) to get different values of y

For x = - 5, y = \frac{15 + 15}{5} = 6 For x = 0, y = \frac{15 - 0}{5} = 3 For x = 5, y = \frac{15 - 15}{5} = 0

So, the table for the second equation (3x + 5y − 15 = 0 ) is

x	−5	0	5
y	6	3	0

Now, plot the points D(−5,6), E(0,3) and F(5,0) on the same graph paper and join
D, E and F to get the graph of 3x + 5y − 15 = 0.

From the graph it is clear that, the given lines intersect at (5,0).
So, the solution of the given system of equations is (5,0).
From the graph, the vertices of the triangle formed by the given lines and the y-axis are
(0,3), (0, −5) and (5,0).
Now,

Area (∆ C E A) = \frac{1}{2} \times E A \times O C = \frac{1}{2} \times 8 \times 5 = 20 sq . units

Hence, the vertices of the triangle formed by the given lines and the y-axis are
(0,3), (0, −5) and (5,0) and its area is 20 sq. units.

Question 19:

From the first equation, write y in terms of x
$y = x - 5 . . . . . (i)$
Substitute different values of x in (i) to get different values of y
$For x = 0, y = 0 - 5 = - 5 For x = 2, y = 2 - 5 = - 3 For x = 5, y = 5 - 5 = 0$
Thus, the table for the first equation (x − y − 5 = 0) is

x	0	2	5
y	−5	−3	0

Now, plot the points A(0,−5), B(2,−3) and C(5,0) on a graph paper and join
A, B and C to get the graph of x − y − 5 = 0.
From the second equation, write y in terms of x

y = \frac{15 - 3 x}{5} . . . . . (i i)

Now, substitute different values of x in (ii) to get different values of y

For x = - 5, y = \frac{15 + 15}{5} = 6 For x = 0, y = \frac{15 - 0}{5} = 3 For x = 5, y = \frac{15 - 15}{5} = 0

So, the table for the second equation (3x + 5y − 15 = 0 ) is

x	−5	0	5
y	6	3	0

Now, plot the points D(−5,6), E(0,3) and F(5,0) on the same graph paper and join
D, E and F to get the graph of 3x + 5y − 15 = 0.

From the graph it is clear that, the given lines intersect at (5,0).
So, the solution of the given system of equations is (5,0).
From the graph, the vertices of the triangle formed by the given lines and the y-axis are
(0,3), (0, −5) and (5,0).
Now,

Area (∆ C E A) = \frac{1}{2} \times E A \times O C = \frac{1}{2} \times 8 \times 5 = 20 sq . units

Hence, the vertices of the triangle formed by the given lines and the y-axis are
(0,3), (0, −5) and (5,0) and its area is 20 sq. units.

Answer:

From the first equation, write y in terms of x
$y = \frac{2 x + 4}{5} . . . . . (i)$
Substitute different values of x in (i) to get different values of y
$For x = - 2, y = \frac{- 4 + 4}{5} = 0 For x = 0, y = \frac{0 + 4}{5} = \frac{4}{5} For x = 3, y = \frac{6 + 4}{5} = 2$
Thus, the table for the first equation (2x − 5y + 4 = 0) is

x	−2	0	3
y	0	$\frac{4}{5}$	2

Now, plot the points A(−2,0),

B (0, \frac{4}{5})

and C(3,2) on a graph paper and join
A, B and C to get the graph of 2x − 5y + 4 = 0.
From the second equation, write y in terms of x

y = 8 - 2 x . . . . . (i i)

Now, substitute different values of x in (ii) to get different values of y

For x = 0, y = 8 - 0 = 8 For x = 2, y = 8 - 4 = 4 For x = 4, y = 8 - 8 = 0

So, the table for the second equation (2x + y − 8 = 0 ) is

x	0	2	4
y	8	4	0

Now, plot the points D(0,8), E(2,4) and F(4,0) on the same graph paper and join
D, E and F to get the graph of 2x + y − 8 = 0.

From the graph it is clear that, the given lines intersect at (3,2).
So, the solution of the given system of equations is (3,2).
The vertices of the triangle formed by the system of equations and y-axis are (0,8),

(0, \frac{4}{5})

and (3,2).
Draw a perpendicular from point C to the y-axis. So,

Area (∆ D B C) = \frac{1}{2} \times D B \times C M = \frac{1}{2} \times (8 - \frac{4}{5}) \times 3 = \frac{54}{5} sq . units

Hence, the veritices of the triangle are (0,8),

(0, \frac{4}{5})

and (3,2) and its area is

\frac{54}{5}

sq. units.

Question 20:

From the first equation, write y in terms of x
$y = \frac{2 x + 4}{5} . . . . . (i)$
Substitute different values of x in (i) to get different values of y
$For x = - 2, y = \frac{- 4 + 4}{5} = 0 For x = 0, y = \frac{0 + 4}{5} = \frac{4}{5} For x = 3, y = \frac{6 + 4}{5} = 2$
Thus, the table for the first equation (2x − 5y + 4 = 0) is

x	−2	0	3
y	0	$\frac{4}{5}$	2

Now, plot the points A(−2,0),

B (0, \frac{4}{5})

and C(3,2) on a graph paper and join
A, B and C to get the graph of 2x − 5y + 4 = 0.
From the second equation, write y in terms of x

y = 8 - 2 x . . . . . (i i)

Now, substitute different values of x in (ii) to get different values of y

For x = 0, y = 8 - 0 = 8 For x = 2, y = 8 - 4 = 4 For x = 4, y = 8 - 8 = 0

So, the table for the second equation (2x + y − 8 = 0 ) is

x	0	2	4
y	8	4	0

Now, plot the points D(0,8), E(2,4) and F(4,0) on the same graph paper and join
D, E and F to get the graph of 2x + y − 8 = 0.

From the graph it is clear that, the given lines intersect at (3,2).
So, the solution of the given system of equations is (3,2).
The vertices of the triangle formed by the system of equations and y-axis are (0,8),

(0, \frac{4}{5})

and (3,2).
Draw a perpendicular from point C to the y-axis. So,

Area (∆ D B C) = \frac{1}{2} \times D B \times C M = \frac{1}{2} \times (8 - \frac{4}{5}) \times 3 = \frac{54}{5} sq . units

Hence, the veritices of the triangle are (0,8),

(0, \frac{4}{5})

and (3,2) and its area is

\frac{54}{5}

sq. units.

Answer:

On a graph paper, draw a horizontal line X'OX and a vertical line YOY' as the x-axis and y-axis, respectively.
Graph of 5x − y = 7
5x − y = 7
⇒ y = (5x − 7) .........(i)
Putting x = 0, we get y = −7.
Putting x = 1, we get y = −2.
Putting x = 2, we get y = 3.
Thus, we have the following table for the equation 5x − y = 7.

x	0	1	2
y	−7	−2	3

Now, plot the points A(0, −7) , B( 1, −2) and C(2, 3) on the graph paper.
Join AB and BC to get the graph line AC. Extend it on both ways.
Thus, AC is the graph of 5x − y = 7.

Graph of x − y + 1 = 0
x − y + 1 = 0
⇒ y = (x + 1) ...........(ii)
Putting x = 0, we get y = 1.
Putting x = 1, we get y = 2.
Putting x = 2, we get y = 3.
Thus, we have the following table for the equation x − y + 1 = 0.

x	0	1	2
y	1	2	3

Now, plot the points P(0, 1) and Q(1, 2). The point C(2, 3) has already been plotted. Join PQ and QC to get the graph line PC. Extend it on both ways.
Then, PC is the graph of the equation x − y + 1 = 0.

The two graph lines intersect at C(2, 3).
∴ The solution of the given system of equations is x = 2 and y = 3.
Clearly, the vertices of ΔAPC formed by these two lines and the y-axis are P(0, 1), C(2, 3) and A(0, −7).
Now, consider ΔAPC.
Here, height = 2 units and base (AP) = 8 units
∴ Area of ΔAPC =

\frac{1}{2} \times base \times height

sq. units
=

(\frac{1}{2} \times 8 \times 2) = 8

sq. units.

Question 21:

On a graph paper, draw a horizontal line X'OX and a vertical line YOY' as the x-axis and y-axis, respectively.
Graph of 5x − y = 7
5x − y = 7
⇒ y = (5x − 7) .........(i)
Putting x = 0, we get y = −7.
Putting x = 1, we get y = −2.
Putting x = 2, we get y = 3.
Thus, we have the following table for the equation 5x − y = 7.

x	0	1	2
y	−7	−2	3

Now, plot the points A(0, −7) , B( 1, −2) and C(2, 3) on the graph paper.
Join AB and BC to get the graph line AC. Extend it on both ways.
Thus, AC is the graph of 5x − y = 7.

Graph of x − y + 1 = 0
x − y + 1 = 0
⇒ y = (x + 1) ...........(ii)
Putting x = 0, we get y = 1.
Putting x = 1, we get y = 2.
Putting x = 2, we get y = 3.
Thus, we have the following table for the equation x − y + 1 = 0.

x	0	1	2
y	1	2	3

Now, plot the points P(0, 1) and Q(1, 2). The point C(2, 3) has already been plotted. Join PQ and QC to get the graph line PC. Extend it on both ways.
Then, PC is the graph of the equation x − y + 1 = 0.

The two graph lines intersect at C(2, 3).
∴ The solution of the given system of equations is x = 2 and y = 3.
Clearly, the vertices of ΔAPC formed by these two lines and the y-axis are P(0, 1), C(2, 3) and A(0, −7).
Now, consider ΔAPC.
Here, height = 2 units and base (AP) = 8 units
∴ Area of ΔAPC =

\frac{1}{2} \times base \times height

sq. units
=

(\frac{1}{2} \times 8 \times 2) = 8

sq. units.

Answer:

From the first equation, write y in terms of x
$y = \frac{2 x - 12}{3} . . . . . (i)$
Substitute different values of x in (i) to get different values of y
$For x = 0, y = \frac{0 - 12}{3} = - 4 For x = 3, y = \frac{6 - 12}{3} = - 2 For x = 6, y = \frac{12 - 12}{3} = 0$
Thus, the table for the first equation (2x − 3y = 12) is

x	0	3	6
y	−4	−2	0

Now, plot the points A(0,−4), B(3,−2) and C(6,0) on a graph paper and join
A, B and C to get the graph of 2x − 3y = 12.
From the second equation, write y in terms of x

y = \frac{6 - x}{3} . . . . . (i i)

Now, substitute different values of x in (ii) to get different values of y

For x = 0, y = \frac{6 - 0}{3} = 2 For x = 3, y = \frac{6 - 3}{3} = 1 For x = 6, y = \frac{6 - 6}{3} = 0

So, the table for the second equation (x + 3y = 6 ) is

x	0	3	6
y	2	1	0

Now, plot the points D(0,2), E(3,1) and F(6,0) on the same graph paper and join
D, E and F to get the graph of x + 3y = 6.

From the graph it is clear that, the given lines intersect at (6,0).
So, the solution of the given system of equations is (6,0).
The vertices of the triangle formed by the system of equations and y-axis are (0,2), (6,0) and (0,−4).
Now,

Area (∆ D A C) = \frac{1}{2} \times D A \times O C = \frac{1}{2} \times 6 \times 6 = 18 sq . units

Hence, the veritices of the triangle are (0,2), (6,0) and (0,−4) and its area is 18 sq. units.

Question 22:

From the first equation, write y in terms of x
$y = \frac{2 x - 12}{3} . . . . . (i)$
Substitute different values of x in (i) to get different values of y
$For x = 0, y = \frac{0 - 12}{3} = - 4 For x = 3, y = \frac{6 - 12}{3} = - 2 For x = 6, y = \frac{12 - 12}{3} = 0$
Thus, the table for the first equation (2x − 3y = 12) is

x	0	3	6
y	−4	−2	0

Now, plot the points A(0,−4), B(3,−2) and C(6,0) on a graph paper and join
A, B and C to get the graph of 2x − 3y = 12.
From the second equation, write y in terms of x

y = \frac{6 - x}{3} . . . . . (i i)

Now, substitute different values of x in (ii) to get different values of y

For x = 0, y = \frac{6 - 0}{3} = 2 For x = 3, y = \frac{6 - 3}{3} = 1 For x = 6, y = \frac{6 - 6}{3} = 0

So, the table for the second equation (x + 3y = 6 ) is

x	0	3	6
y	2	1	0

Now, plot the points D(0,2), E(3,1) and F(6,0) on the same graph paper and join
D, E and F to get the graph of x + 3y = 6.

From the graph it is clear that, the given lines intersect at (6,0).
So, the solution of the given system of equations is (6,0).
The vertices of the triangle formed by the system of equations and y-axis are (0,2), (6,0) and (0,−4).
Now,

Area (∆ D A C) = \frac{1}{2} \times D A \times O C = \frac{1}{2} \times 6 \times 6 = 18 sq . units

Hence, the veritices of the triangle are (0,2), (6,0) and (0,−4) and its area is 18 sq. units.

Answer:

From the first equation, write y in terms of x
$y = \frac{6 - 2 x}{3} . . . . . (i)$
Substitute different values of x in (i) to get different values of y
$For x = - 3, y = \frac{6 + 6}{3} = 4 For x = 3, y = \frac{6 - 6}{3} = 0 For x = 6, y = \frac{6 - 12}{3} = - 2$
Thus, the table for the first equation (2x + 3y = 6) is

x	−3	3	6
y	4	0	−2

Now, plot the points A(−3,4), B(3,0) and C(6,−2) on a graph paper and join
A, B and C to get the graph of 2x + 3y = 6.
From the second equation, write y in terms of x

y = \frac{12 - 4 x}{6} . . . . . (i i)

Now, substitute different values of x in (ii) to get different values of y

For x = - 6, y = \frac{12 + 24}{6} = 6 For x = 0, y = \frac{12 - 0}{6} = 2 For x = 9, y = \frac{12 - 36}{6} = - 4

So, the table for the second equation (4x + 6y = 12 ) is

x	−6	0	9
y	6	2	−4

Now, plot the points D(−6,6), E(0,2) and F(9,−4) on the same graph paper and join
D, E and F to get the graph of 4x + 6y = 12.

From the graph it is clear that, the given lines coincide with each other.
Hence, the solution of the given system of equations has infinitely many solutions.

Question 23:

From the first equation, write y in terms of x
$y = \frac{6 - 2 x}{3} . . . . . (i)$
Substitute different values of x in (i) to get different values of y
$For x = - 3, y = \frac{6 + 6}{3} = 4 For x = 3, y = \frac{6 - 6}{3} = 0 For x = 6, y = \frac{6 - 12}{3} = - 2$
Thus, the table for the first equation (2x + 3y = 6) is

x	−3	3	6
y	4	0	−2

Now, plot the points A(−3,4), B(3,0) and C(6,−2) on a graph paper and join
A, B and C to get the graph of 2x + 3y = 6.
From the second equation, write y in terms of x

y = \frac{12 - 4 x}{6} . . . . . (i i)

Now, substitute different values of x in (ii) to get different values of y

For x = - 6, y = \frac{12 + 24}{6} = 6 For x = 0, y = \frac{12 - 0}{6} = 2 For x = 9, y = \frac{12 - 36}{6} = - 4

So, the table for the second equation (4x + 6y = 12 ) is

x	−6	0	9
y	6	2	−4

Now, plot the points D(−6,6), E(0,2) and F(9,−4) on the same graph paper and join
D, E and F to get the graph of 4x + 6y = 12.

From the graph it is clear that, the given lines coincide with each other.
Hence, the solution of the given system of equations has infinitely many solutions.

Answer:

On a graph paper, draw a horizontal line X'OX and a vertical line YOY' as the x-axis and y-axis, respectively.
Graph of 3x − y = 5
3x − y = 5
⇒ y = (3x − 5) .........(i)
Putting x = 1, we get y = −2.
Putting x = 0, we get y = −5.
Putting x = 2, we get y = 1.
Thus, we have the following table for the equation 3x − y = 5.

x	1	0	2
y	−2	−5	1

Now, plot the points A(1, −2), B(0, −5) and C(2, 1) on the graph paper.
Join AB and AC to get the graph line BC. Extend it on both ways.
Thus, BC is the graph of 3x − y = 5.

Graph of 6x − 2y = 10
6x − 2y = 10
⇒ 2y = (6x − 10)
∴

y = \frac{6 x - 10}{2}

...........(ii)
Putting x = 0, we get y = −5.
Putting x = 1, we get y = −2.
Putting x = 2, we get y = 1.

Thus, we have the following table for the equation 6x − 2y = 10.

x	0	1	2
y	−5	−2	1

These are the same points as obtained for the graph line of equation (i).

It is clear from the graph that these two lines coincide.
Hence, the given system of equations has infinitely many solutions.

Question 24:

On a graph paper, draw a horizontal line X'OX and a vertical line YOY' as the x-axis and y-axis, respectively.
Graph of 3x − y = 5
3x − y = 5
⇒ y = (3x − 5) .........(i)
Putting x = 1, we get y = −2.
Putting x = 0, we get y = −5.
Putting x = 2, we get y = 1.
Thus, we have the following table for the equation 3x − y = 5.

x	1	0	2
y	−2	−5	1

Now, plot the points A(1, −2), B(0, −5) and C(2, 1) on the graph paper.
Join AB and AC to get the graph line BC. Extend it on both ways.
Thus, BC is the graph of 3x − y = 5.

Graph of 6x − 2y = 10
6x − 2y = 10
⇒ 2y = (6x − 10)
∴

y = \frac{6 x - 10}{2}

...........(ii)
Putting x = 0, we get y = −5.
Putting x = 1, we get y = −2.
Putting x = 2, we get y = 1.

Thus, we have the following table for the equation 6x − 2y = 10.

x	0	1	2
y	−5	−2	1

These are the same points as obtained for the graph line of equation (i).

It is clear from the graph that these two lines coincide.
Hence, the given system of equations has infinitely many solutions.

Answer:

On a graph paper, draw a horizontal line X'OX and a vertical line YOY' as the x-axis and y-axis, respectively.
Graph of 2x + y = 6
2x + y = 6 ⇒ y = (6 − 2x) ...(i)
Putting x = 3, we get y = 0.
Putting x = 1, we get y = 4.
Putting x = 2, we get y = 2.
Thus, we have the following table for the equation 2x + y = 6.

x	3	1	2
y	0	4	2

Now, plot the points A(3, 0), B(1, 4) and C(2, 2) on the graph paper.
Join AC and CB to get the graph line AB. Extend it on both ways.
Thus, AB is the graph of 2x + y = 6.

Graph of 6x + 3y = 18
6x + 3y = 18 ⇒ 3y = (18 − 6x)
∴

y = \frac{18 - 6 x}{3}

...........(ii)
Putting x = 3, we get y = 0.
Putting x = 1, we get y = 4.
Putting x = 2, we get y = 2.
Thus, we have the following table for the equation 6x + 3y = 18.

x	3	1	2
y	0	4	2

These are the same points as obtained for the graph line of equation (i).

It is clear from the graph that these two lines coincide.
Hence, the given system of equations has an infinite number of solutions.

Question 25:

On a graph paper, draw a horizontal line X'OX and a vertical line YOY' as the x-axis and y-axis, respectively.
Graph of 2x + y = 6
2x + y = 6 ⇒ y = (6 − 2x) ...(i)
Putting x = 3, we get y = 0.
Putting x = 1, we get y = 4.
Putting x = 2, we get y = 2.
Thus, we have the following table for the equation 2x + y = 6.

x	3	1	2
y	0	4	2

Now, plot the points A(3, 0), B(1, 4) and C(2, 2) on the graph paper.
Join AC and CB to get the graph line AB. Extend it on both ways.
Thus, AB is the graph of 2x + y = 6.

Graph of 6x + 3y = 18
6x + 3y = 18 ⇒ 3y = (18 − 6x)
∴

y = \frac{18 - 6 x}{3}

...........(ii)
Putting x = 3, we get y = 0.
Putting x = 1, we get y = 4.
Putting x = 2, we get y = 2.
Thus, we have the following table for the equation 6x + 3y = 18.

x	3	1	2
y	0	4	2

These are the same points as obtained for the graph line of equation (i).

It is clear from the graph that these two lines coincide.
Hence, the given system of equations has an infinite number of solutions.

Answer:

From the first equation, write y in terms of x
$y = \frac{x - 5}{2} . . . . . (i)$
Substitute different values of x in (i) to get different values of y
$For x = - 5, y = \frac{- 5 - 5}{2} = - 5 For x = 1, y = \frac{1 - 5}{2} = - 2 For x = 3, y = \frac{3 - 5}{2} = - 1$
Thus, the table for the first equation (x − 2y = 5) is

x	−5	1	3
y	−5	−2	−1

Now, plot the points A(−5,−5), B(1,−2) and C(3,−1) on a graph paper and join
A, B and C to get the graph of x − 2y = 5.
From the second equation, write y in terms of x

y = \frac{3 x - 15}{6} . . . . . (i i)

Now, substitute different values of x in (ii) to get different values of y

For x = - 3, y = \frac{- 9 - 15}{6} = - 4 For x = - 1, y = \frac{- 3 - 15}{6} = - 3 For x = 5, y = \frac{15 - 15}{6} = 0

So, the table for the second equation (3x − 6y = 15 ) is

x	−3	−1	5
y	−4	−3	0

Now, plot the points D(−3,−4), E(−1,−3) and F(5,0) on the same graph paper and join
D, E and F to get the graph of 3x − 6y = 15.

From the graph it is clear that, the given lines coincide with each other.
Hence, the solution of the given system of equations has infinitely many solutions.

Question 26:

From the first equation, write y in terms of x
$y = \frac{x - 5}{2} . . . . . (i)$
Substitute different values of x in (i) to get different values of y
$For x = - 5, y = \frac{- 5 - 5}{2} = - 5 For x = 1, y = \frac{1 - 5}{2} = - 2 For x = 3, y = \frac{3 - 5}{2} = - 1$
Thus, the table for the first equation (x − 2y = 5) is

x	−5	1	3
y	−5	−2	−1

Now, plot the points A(−5,−5), B(1,−2) and C(3,−1) on a graph paper and join
A, B and C to get the graph of x − 2y = 5.
From the second equation, write y in terms of x

y = \frac{3 x - 15}{6} . . . . . (i i)

Now, substitute different values of x in (ii) to get different values of y

For x = - 3, y = \frac{- 9 - 15}{6} = - 4 For x = - 1, y = \frac{- 3 - 15}{6} = - 3 For x = 5, y = \frac{15 - 15}{6} = 0

So, the table for the second equation (3x − 6y = 15 ) is

x	−3	−1	5
y	−4	−3	0

Now, plot the points D(−3,−4), E(−1,−3) and F(5,0) on the same graph paper and join
D, E and F to get the graph of 3x − 6y = 15.

From the graph it is clear that, the given lines coincide with each other.
Hence, the solution of the given system of equations has infinitely many solutions.

Answer:

From the first equation, write y in terms of x
$y = \frac{x - 6}{2} . . . . . (i)$
Substitute different values of x in (i) to get different values of y
$For x = - 2, y = \frac{- 2 - 6}{2} = - 4 For x = 0, y = \frac{0 - 6}{2} = - 3 For x = 2, y = \frac{2 - 6}{2} = - 2$
Thus, the table for the first equation (x − 2y = 6) is

x	−2	0	2
y	−4	−3	−2

Now, plot the points A(−2,−4), B(0,−3) and C(2,−2) on a graph paper and join
A, B and C to get the graph of x − 2y = 6.
From the second equation, write y in terms of x

y = \frac{1}{2} x . . . . . (i i)

Now, substitute different values of x in (ii) to get different values of y

For x = - 4, y = \frac{- 4}{2} = - 2 For x = 0, y = \frac{0}{2} = 0 For x = 4, y = \frac{4}{2} = 2

So, the table for the second equation (3x − 6y = 0 ) is

x	−4	0	4
y	−2	0	2

Now, plot the points D(−4,−2), O(0,0) and E(4,2) on the same graph paper and join
D, E and F to get the graph of 3x − 6y = 0.

From the graph it is clear that, the given lines do not intersect at all when produced.
Hence, the system of equations has no solution and therefore is inconsistent.

Question 27:

From the first equation, write y in terms of x
$y = \frac{x - 6}{2} . . . . . (i)$
Substitute different values of x in (i) to get different values of y
$For x = - 2, y = \frac{- 2 - 6}{2} = - 4 For x = 0, y = \frac{0 - 6}{2} = - 3 For x = 2, y = \frac{2 - 6}{2} = - 2$
Thus, the table for the first equation (x − 2y = 6) is

x	−2	0	2
y	−4	−3	−2

Now, plot the points A(−2,−4), B(0,−3) and C(2,−2) on a graph paper and join
A, B and C to get the graph of x − 2y = 6.
From the second equation, write y in terms of x

y = \frac{1}{2} x . . . . . (i i)

Now, substitute different values of x in (ii) to get different values of y

For x = - 4, y = \frac{- 4}{2} = - 2 For x = 0, y = \frac{0}{2} = 0 For x = 4, y = \frac{4}{2} = 2

So, the table for the second equation (3x − 6y = 0 ) is

x	−4	0	4
y	−2	0	2

Now, plot the points D(−4,−2), O(0,0) and E(4,2) on the same graph paper and join
D, E and F to get the graph of 3x − 6y = 0.

From the graph it is clear that, the given lines do not intersect at all when produced.
Hence, the system of equations has no solution and therefore is inconsistent.

Answer:

On a graph paper, draw a horizontal line X'OX and a vertical line YOY' as the x-axis and y-axis, respectively.
Graph of 2x + 3y = 4
2x + 3y = 4
⇒ 3y = (−2x + 4)
∴ $y = \frac{- 2 x + 4}{3}$ .........(i)
Putting x = 2, we get y = 0.
Putting x = −1, we get y = 2.
Putting x = −4, we get y = 4.
Thus, we have the following table for the equation 2x + 3y = 4.

x	2	−1	−4
y	0	2	4

Now, plot the points A(2, 0), B(−1, 2) and C(−4, 4) on the graph paper.
Join AB and BC to get the graph line AC. Extend it on both ways.
Thus, AC is the graph of 2x + 3y = 4.

Graph of 4x + 6y = 12
4x + 6y = 12
⇒ 6y = (−4x + 12)
∴

y = \frac{- 4 x + 12}{6}

...........(ii)
Putting x = 3, we get y = 0.
Putting x = 0, we get y = 2.
Putting x = 6, we get y = −2.
Thus, we have the following table for the equation 4x + 6y = 12.

x	3	0	6
y	0	2	−2

Now, on the same graph, plot the points P(3, 0), Q(0, 2) and R (6, −2).
Join PQ and PR to get the graph line QR. Extend it on both ways.
Then, QR is the graph of the equation 4x + 6y = 12.

It is clear from the graph that these two lines are parallel and do not intersect when produced.
Hence, the given system of equations is inconsistent.

Question 28:

On a graph paper, draw a horizontal line X'OX and a vertical line YOY' as the x-axis and y-axis, respectively.
Graph of 2x + 3y = 4
2x + 3y = 4
⇒ 3y = (−2x + 4)
∴ $y = \frac{- 2 x + 4}{3}$ .........(i)
Putting x = 2, we get y = 0.
Putting x = −1, we get y = 2.
Putting x = −4, we get y = 4.
Thus, we have the following table for the equation 2x + 3y = 4.

x	2	−1	−4
y	0	2	4

Now, plot the points A(2, 0), B(−1, 2) and C(−4, 4) on the graph paper.
Join AB and BC to get the graph line AC. Extend it on both ways.
Thus, AC is the graph of 2x + 3y = 4.

Graph of 4x + 6y = 12
4x + 6y = 12
⇒ 6y = (−4x + 12)
∴

y = \frac{- 4 x + 12}{6}

...........(ii)
Putting x = 3, we get y = 0.
Putting x = 0, we get y = 2.
Putting x = 6, we get y = −2.
Thus, we have the following table for the equation 4x + 6y = 12.

x	3	0	6
y	0	2	−2

Now, on the same graph, plot the points P(3, 0), Q(0, 2) and R (6, −2).
Join PQ and PR to get the graph line QR. Extend it on both ways.
Then, QR is the graph of the equation 4x + 6y = 12.

It is clear from the graph that these two lines are parallel and do not intersect when produced.
Hence, the given system of equations is inconsistent.

Answer:

From the first equation, write y in terms of x
$y = 6 - 2 x . . . . . (i)$
Substitute different values of x in (i) to get different values of y
$For x = 0, y = 6 - 0 = 6 For x = 2, y = 6 - 4 = 2 For x = 4, y = 6 - 8 = - 2$
Thus, the table for the first equation (2x + y = 6) is

x	0	2	4
y	6	2	−2

Now, plot the points A(0,6), B(2,2) and C(4,−2) on a graph paper and join
A, B and C to get the graph of 2x + y = 6.
From the second equation, write y in terms of x

y = \frac{20 - 6 x}{3} . . . . . (i i)

Now, substitute different values of x in (ii) to get different values of y

For x = 0, y = \frac{20 - 0}{3} = \frac{20}{3} For x = \frac{10}{3}, y = \frac{20 - 20}{3} = 0 For x = 5, y = \frac{20 - 30}{3} = - \frac{10}{3}

So, the table for the second equation (6x + 3y = 20 ) is

x	0	$\frac{10}{3}$	5
y	$\frac{20}{3}$	0	$- \frac{10}{3}$

Now, plot the points

D (0, \frac{20}{3}), E (\frac{10}{3}, 0) and F (5, - \frac{10}{3})

on the same graph paper and join
D, E and F to get the graph of 6x + 3y = 20.

From the graph it is clear that, the given lines do not intersect at all when produced.
Hence, the system of equations has no solution and therefore is inconsistent.

Question 29:

From the first equation, write y in terms of x
$y = 6 - 2 x . . . . . (i)$
Substitute different values of x in (i) to get different values of y
$For x = 0, y = 6 - 0 = 6 For x = 2, y = 6 - 4 = 2 For x = 4, y = 6 - 8 = - 2$
Thus, the table for the first equation (2x + y = 6) is

x	0	2	4
y	6	2	−2

Now, plot the points A(0,6), B(2,2) and C(4,−2) on a graph paper and join
A, B and C to get the graph of 2x + y = 6.
From the second equation, write y in terms of x

y = \frac{20 - 6 x}{3} . . . . . (i i)

Now, substitute different values of x in (ii) to get different values of y

For x = 0, y = \frac{20 - 0}{3} = \frac{20}{3} For x = \frac{10}{3}, y = \frac{20 - 20}{3} = 0 For x = 5, y = \frac{20 - 30}{3} = - \frac{10}{3}

So, the table for the second equation (6x + 3y = 20 ) is

x	0	$\frac{10}{3}$	5
y	$\frac{20}{3}$	0	$- \frac{10}{3}$

Now, plot the points

D (0, \frac{20}{3}), E (\frac{10}{3}, 0) and F (5, - \frac{10}{3})

on the same graph paper and join
D, E and F to get the graph of 6x + 3y = 20.

From the graph it is clear that, the given lines do not intersect at all when produced.
Hence, the system of equations has no solution and therefore is inconsistent.

Answer:

From the first equation, write y in terms of x
$y = 2 - 2 x . . . . . (i)$
Substitute different values of x in (i) to get different values of y
$For x = 0, y = 2 - 0 = 2 For x = 1, y = 2 - 2 = 0 For x = 2, y = 2 - 4 = - 2$
Thus, the table for the first equation (2x + y = 2) is

x	0	1	2
y	2	0	−2

Now, plot the points A(0,2), B(1,0) and C(2,−2) on a graph paper and join
A, B and C to get the graph of 2x + y = 2.
From the second equation, write y in terms of x

y = 6 - 2 x . . . . . (i i)

Now, substitute different values of x in (ii) to get different values of y

For x = 0, y = 6 - 0 = 6 For x = 1, y = 6 - 2 = 4 For x = 3, y = 6 - 6 = 0

So, the table for the second equation (2x + y = 6 ) is

x	0	1	3
y	6	4	0

Now, plot the points D(0,6), E(1,4) and F(3,0)on the same graph paper and join
D, E and F to get the graph of 2x + y = 6.

From the graph it is clear that, the given lines do not intersect at all when produced. So, these lines are
parallel to each other and therefore, the quadrilateral DABF is a trapezium. The vertices of the
required trapezium are D(0,6), A(0,2), B(1,0) and F(3,0).
Now,

Area (T r a p e z i u m D A B F) = Area (∆ D O F) - Area (∆ A O B) = \frac{1}{2} \times 3 \times 6 - \frac{1}{2} \times 1 \times 2 = 9 - 1 = 8 sq . units

Hence, the area of the rquired trapezium is 8 sq. units.

Question 1:

From the first equation, write y in terms of x
$y = 2 - 2 x . . . . . (i)$
Substitute different values of x in (i) to get different values of y
$For x = 0, y = 2 - 0 = 2 For x = 1, y = 2 - 2 = 0 For x = 2, y = 2 - 4 = - 2$
Thus, the table for the first equation (2x + y = 2) is

x	0	1	2
y	2	0	−2

Now, plot the points A(0,2), B(1,0) and C(2,−2) on a graph paper and join
A, B and C to get the graph of 2x + y = 2.
From the second equation, write y in terms of x

y = 6 - 2 x . . . . . (i i)

Now, substitute different values of x in (ii) to get different values of y

For x = 0, y = 6 - 0 = 6 For x = 1, y = 6 - 2 = 4 For x = 3, y = 6 - 6 = 0

So, the table for the second equation (2x + y = 6 ) is

x	0	1	3
y	6	4	0

Now, plot the points D(0,6), E(1,4) and F(3,0)on the same graph paper and join
D, E and F to get the graph of 2x + y = 6.

From the graph it is clear that, the given lines do not intersect at all when produced. So, these lines are
parallel to each other and therefore, the quadrilateral DABF is a trapezium. The vertices of the
required trapezium are D(0,6), A(0,2), B(1,0) and F(3,0).
Now,

Area (T r a p e z i u m D A B F) = Area (∆ D O F) - Area (∆ A O B) = \frac{1}{2} \times 3 \times 6 - \frac{1}{2} \times 1 \times 2 = 9 - 1 = 8 sq . units

Hence, the area of the rquired trapezium is 8 sq. units.

Answer:

The given system of equation is:
x + y = 3 .........(i)
4x − 3y = 26 ........(ii)

On multiplying (i) by 3, we get:
3x + 3y = 9 .......(iii)

On adding (ii) and (iii), we get:
7x = 35
⇒ x = 5

On substituting the value of x = 5 in (i), we get:
5 + y = 3
⇒ y = (3 − 5) = −2

Hence, the solution is x = 5 and y = −2

Question 2:

The given system of equation is:
x + y = 3 .........(i)
4x − 3y = 26 ........(ii)

On multiplying (i) by 3, we get:
3x + 3y = 9 .......(iii)

On adding (ii) and (iii), we get:
7x = 35
⇒ x = 5

On substituting the value of x = 5 in (i), we get:
5 + y = 3
⇒ y = (3 − 5) = −2

Hence, the solution is x = 5 and y = −2

Answer:

The given system of equations is
$x - y = 3 . . . . . (i) \frac{x}{3} + \frac{y}{2} = 6 . . . . . (i i)$
From (i), write y in terms of x to get
$y = x - 3$
Substituting y = x − 3 in (ii), we get
$\frac{x}{3} + \frac{x - 3}{2} = 6 \Rightarrow 2 x + 3 (x - 3) = 36 \Rightarrow 2 x + 3 x - 9 = 36 \Rightarrow x = \frac{45}{5} = 9$
Now, substituting x = 9 in (i), we have
$9 - y = 3 \Rightarrow y = 9 - 3 = 6$
Hence, x = 9 and y = 6.

Question 3:

The given system of equations is
$x - y = 3 . . . . . (i) \frac{x}{3} + \frac{y}{2} = 6 . . . . . (i i)$
From (i), write y in terms of x to get
$y = x - 3$
Substituting y = x − 3 in (ii), we get
$\frac{x}{3} + \frac{x - 3}{2} = 6 \Rightarrow 2 x + 3 (x - 3) = 36 \Rightarrow 2 x + 3 x - 9 = 36 \Rightarrow x = \frac{45}{5} = 9$
Now, substituting x = 9 in (i), we have
$9 - y = 3 \Rightarrow y = 9 - 3 = 6$
Hence, x = 9 and y = 6.

Answer:

The given system of equation is:
2x + 3y = 0 .........(i)
3x + 4y = 5 ........(ii)

On multiplying (i) by 4 and (ii) by 3, we get:
8x + 12y = 0 .......(iii)
9x + 12y = 15 ......(iv)

On subtracting (iii) from (iv) we get:
x = 15

On substituting the value of x = 15 in (i), we get:
30 + 3y = 0
⇒ 3y = −30
⇒ y = −10

Hence, the solution is x = 15 and y = −10.

Question 4:

The given system of equation is:
2x + 3y = 0 .........(i)
3x + 4y = 5 ........(ii)

On multiplying (i) by 4 and (ii) by 3, we get:
8x + 12y = 0 .......(iii)
9x + 12y = 15 ......(iv)

On subtracting (iii) from (iv) we get:
x = 15

On substituting the value of x = 15 in (i), we get:
30 + 3y = 0
⇒ 3y = −30
⇒ y = −10

Hence, the solution is x = 15 and y = −10.

Answer:

The given system of equation is:
2x − 3y = 13 .........(i)
7x − 2y = 20 ........(ii)

On multiplying (i) by 2 and (ii) by 3, we get:
4x − 6y = 26 .......(iii)
21x − 6y = 60 ......(iv)

On subtracting (iii) from (iv) we get:
17x = (60 − 26) = 34
⇒ x = 2

On substituting the value of x = 2 in (i), we get:
4 − 3y = 13
⇒ 3y = (4 − 13) = −9
⇒ y = −3

Hence, the solution is x = 2 and y = −3

Question 5:

The given system of equation is:
2x − 3y = 13 .........(i)
7x − 2y = 20 ........(ii)

On multiplying (i) by 2 and (ii) by 3, we get:
4x − 6y = 26 .......(iii)
21x − 6y = 60 ......(iv)

On subtracting (iii) from (iv) we get:
17x = (60 − 26) = 34
⇒ x = 2

On substituting the value of x = 2 in (i), we get:
4 − 3y = 13
⇒ 3y = (4 − 13) = −9
⇒ y = −3

Hence, the solution is x = 2 and y = −3

Answer:

The given system of equation is:
3x − 5y − 19 = 0 .........(i)
−7x + 3y + 1 = 0 ........(ii)

On multiplying (i) by 3 and (ii) by 5, we get:
9x − 15y = 57 or .........(iii)
−35x + 15y = −5 ........(iv)

On adding (iii) from (iv) we get:
−26x = (57 − 5)) = 52
⇒ x = −2

On substituting the value of x = −2 in (i), we get:
−6 − 5y − 19 = 0
⇒ 5y = (− 6 − 19) = −25
⇒ y = −5

Hence, the solution is x = −2 and y = −5.

Question 6:

The given system of equation is:
3x − 5y − 19 = 0 .........(i)
−7x + 3y + 1 = 0 ........(ii)

On multiplying (i) by 3 and (ii) by 5, we get:
9x − 15y = 57 or .........(iii)
−35x + 15y = −5 ........(iv)

On adding (iii) from (iv) we get:
−26x = (57 − 5)) = 52
⇒ x = −2

On substituting the value of x = −2 in (i), we get:
−6 − 5y − 19 = 0
⇒ 5y = (− 6 − 19) = −25
⇒ y = −5

Hence, the solution is x = −2 and y = −5.

Answer:

$Given : 3 x - 5 y = 4 . . . (1) 2 y + 7 = 9 x . . . (2) 2 y + 7 = 9 x \Rightarrow 2 y = 9 x - 7 \Rightarrow y = \frac{9 x - 7}{2} . . . (3) Substituting the value of y in (1), we get 3 x - 5 (\frac{9 x - 7}{2}) = 4 \Rightarrow \frac{6 x - 5 (9 x - 7)}{2} = 4 \Rightarrow 6 x - 45 x + 35 = 8 \Rightarrow - 39 x = 8 - 35 \Rightarrow - 39 x = - 27 \Rightarrow x = \frac{- 27}{- 39} \Rightarrow x = \frac{9}{13} . . . (4) Substituting the value of x in (3), we get y = \frac{9 (\frac{9}{13}) - 7}{2} \Rightarrow y = \frac{81 - 91}{26} \Rightarrow y = \frac{- 10}{26} \Rightarrow y = - \frac{5}{13} Hence, x = \frac{9}{13} and y = - \frac{5}{13} .$

Question 7:

$Given : 3 x - 5 y = 4 . . . (1) 2 y + 7 = 9 x . . . (2) 2 y + 7 = 9 x \Rightarrow 2 y = 9 x - 7 \Rightarrow y = \frac{9 x - 7}{2} . . . (3) Substituting the value of y in (1), we get 3 x - 5 (\frac{9 x - 7}{2}) = 4 \Rightarrow \frac{6 x - 5 (9 x - 7)}{2} = 4 \Rightarrow 6 x - 45 x + 35 = 8 \Rightarrow - 39 x = 8 - 35 \Rightarrow - 39 x = - 27 \Rightarrow x = \frac{- 27}{- 39} \Rightarrow x = \frac{9}{13} . . . (4) Substituting the value of x in (3), we get y = \frac{9 (\frac{9}{13}) - 7}{2} \Rightarrow y = \frac{81 - 91}{26} \Rightarrow y = \frac{- 10}{26} \Rightarrow y = - \frac{5}{13} Hence, x = \frac{9}{13} and y = - \frac{5}{13} .$

Answer:

The given system of equations can be written as
$9 x - 2 y = 108 . . . . . (i) 3 x + 7 y = 105 . . . . . (i i)$
Multiplying (i) by 7 and (ii) by 2, we get
$63 x + 6 x = 108 \times 7 + 105 \times 2 \Rightarrow 69 x = 966 \Rightarrow x = \frac{966}{69} = 14$
Now, substituting x = 14 in (1), we have
$9 \times 14 - 2 y = 108 \Rightarrow 2 y = 126 - 108 \Rightarrow y = \frac{18}{2} = 9$
Hence, x = 14 and y = 9.

Question 8:

The given system of equations can be written as
$9 x - 2 y = 108 . . . . . (i) 3 x + 7 y = 105 . . . . . (i i)$
Multiplying (i) by 7 and (ii) by 2, we get
$63 x + 6 x = 108 \times 7 + 105 \times 2 \Rightarrow 69 x = 966 \Rightarrow x = \frac{966}{69} = 14$
Now, substituting x = 14 in (1), we have
$9 \times 14 - 2 y = 108 \Rightarrow 2 y = 126 - 108 \Rightarrow y = \frac{18}{2} = 9$
Hence, x = 14 and y = 9.

Answer:

The given equations are:
$\frac{x}{3} + \frac{y}{4} = 11$
⇒ 4x + 3y = 132 ........(i)

and $\frac{5 x}{6} - \frac{y}{3} = - 7$
⇒ 5x − 2y = −42..........(ii)

On multiplying (i) by 2 and (ii) by 3, we get:
8x + 6y = 264 ........(iii)
15x − 6y = −126 ...........(iv)

On adding (iii) and (iv), we get:
23x = 138
⇒ x = 6

On substituting x = 6 in (i), we get:
24 + 3y = 132
⇒ 3y = (132 − 24) = 108
⇒ y = 36

Hence, the solution is x = 6 and y = 36.

Question 9:

The given equations are:
$\frac{x}{3} + \frac{y}{4} = 11$
⇒ 4x + 3y = 132 ........(i)

and $\frac{5 x}{6} - \frac{y}{3} = - 7$
⇒ 5x − 2y = −42..........(ii)

On multiplying (i) by 2 and (ii) by 3, we get:
8x + 6y = 264 ........(iii)
15x − 6y = −126 ...........(iv)

On adding (iii) and (iv), we get:
23x = 138
⇒ x = 6

On substituting x = 6 in (i), we get:
24 + 3y = 132
⇒ 3y = (132 − 24) = 108
⇒ y = 36

Hence, the solution is x = 6 and y = 36.

Answer:

The given system of equation is:
4x − 3y = 8 .........(i)
$6 x - y = \frac{29}{3}$ ........(ii)
On multiplying (ii) by 3, we get:
18x − 3y = 29.........(iii)

On subtracting (iii) from (i) we get:
−14x = −21
⇒ x = $\frac{21}{14} = \frac{3}{2}$
On substituting the value of x = $\frac{3}{2}$ in (i), we get:
$4 \times \frac{3}{2} - 3 y = 8 \Rightarrow 6 - 3 y = 8 \Rightarrow 3 y = 6 - 8 = - 2 \Rightarrow y = \frac{- 2}{3}$
Hence, the solution is x = $\frac{3}{2}$ and y = $\frac{- 2}{3}$ .

Question 10:

The given system of equation is:
4x − 3y = 8 .........(i)
$6 x - y = \frac{29}{3}$ ........(ii)
On multiplying (ii) by 3, we get:
18x − 3y = 29.........(iii)

On subtracting (iii) from (i) we get:
−14x = −21
⇒ x = $\frac{21}{14} = \frac{3}{2}$
On substituting the value of x = $\frac{3}{2}$ in (i), we get:
$4 \times \frac{3}{2} - 3 y = 8 \Rightarrow 6 - 3 y = 8 \Rightarrow 3 y = 6 - 8 = - 2 \Rightarrow y = \frac{- 2}{3}$
Hence, the solution is x = $\frac{3}{2}$ and y = $\frac{- 2}{3}$ .

Answer:

The given equations are:
$2 x - \frac{3 y}{4} = 3$ ........(i)
5x = 2y + 7 ............(ii)

On multiplying (i) by 2 and (ii) by $\frac{3}{4}$ , we get:
$4 x - \frac{3}{2} y = 6$ .......(iii)
$\frac{15}{4} x = \frac{3}{2} y + \frac{21}{4}$ .......(iv)

On subtracting (iii) from (iv), we get:
$- \frac{1}{4} x = - \frac{3}{4} \Rightarrow x = 3$

On substituting x = 3 in (i), we get:
$2 \times 3 - \frac{3 y}{4} = 3 \Rightarrow \frac{3 y}{4} = (6 - 3) = 3 \Rightarrow y = \frac{3 \times 4}{3} = 4$

Hence, the solution is x = 3 and y = 4.

Question 11:

The given equations are:
$2 x - \frac{3 y}{4} = 3$ ........(i)
5x = 2y + 7 ............(ii)

On multiplying (i) by 2 and (ii) by $\frac{3}{4}$ , we get:
$4 x - \frac{3}{2} y = 6$ .......(iii)
$\frac{15}{4} x = \frac{3}{2} y + \frac{21}{4}$ .......(iv)

On subtracting (iii) from (iv), we get:
$- \frac{1}{4} x = - \frac{3}{4} \Rightarrow x = 3$

On substituting x = 3 in (i), we get:
$2 \times 3 - \frac{3 y}{4} = 3 \Rightarrow \frac{3 y}{4} = (6 - 3) = 3 \Rightarrow y = \frac{3 \times 4}{3} = 4$

Hence, the solution is x = 3 and y = 4.

Answer:

The given equations are:
$2 x + 5 y = \frac{8}{3}$ ........(i)
$3 x - 2 y = \frac{5}{6}$ ..........(ii)

On multiplying (i) by 2 and (ii) by 5, we get:
$4 x + 10 y = \frac{16}{3}$ ........(iii)
$15 x - 10 y = \frac{25}{6}$ ...........(iv)

On adding (iii) and (iv), we get:
$19 x = \frac{57}{6} \Rightarrow x = \frac{57}{6 \times 19} = \frac{3}{6} = \frac{1}{2}$

On substituting x = $\frac{1}{2}$ in (i), we get:
$2 \times \frac{1}{2} + 5 y = \frac{8}{3}$
$\Rightarrow 5 y = (\frac{8}{3} - 1) = \frac{5}{3} \Rightarrow y = \frac{5}{3 \times 5} = \frac{1}{3}$

Hence, the solution is x = $\frac{1}{2}$ and y = $\frac{1}{3}$ .

Question 12:

The given equations are:
$2 x + 5 y = \frac{8}{3}$ ........(i)
$3 x - 2 y = \frac{5}{6}$ ..........(ii)

On multiplying (i) by 2 and (ii) by 5, we get:
$4 x + 10 y = \frac{16}{3}$ ........(iii)
$15 x - 10 y = \frac{25}{6}$ ...........(iv)

On adding (iii) and (iv), we get:
$19 x = \frac{57}{6} \Rightarrow x = \frac{57}{6 \times 19} = \frac{3}{6} = \frac{1}{2}$

On substituting x = $\frac{1}{2}$ in (i), we get:
$2 \times \frac{1}{2} + 5 y = \frac{8}{3}$
$\Rightarrow 5 y = (\frac{8}{3} - 1) = \frac{5}{3} \Rightarrow y = \frac{5}{3 \times 5} = \frac{1}{3}$

Hence, the solution is x = $\frac{1}{2}$ and y = $\frac{1}{3}$ .

Answer:

The given equations are:
$\frac{7 - 4 x}{3} = y$
⇒ 4x + 3y = 7 .......(i)

and 2x + 3y + 1 = 0
⇒ 2x + 3y = −1 .............(ii)

On subtracting (ii) from (i), we get:
2x = 8
⇒ x = 4

On substituting x = 4 in (i), we get:
16 + 3y = 7
⇒ 3y = (7 − 16) = −9
⇒ y = −3

Hence, the solution is x = 4 and y = −3.

Question 13:

The given equations are:
$\frac{7 - 4 x}{3} = y$
⇒ 4x + 3y = 7 .......(i)

and 2x + 3y + 1 = 0
⇒ 2x + 3y = −1 .............(ii)

On subtracting (ii) from (i), we get:
2x = 8
⇒ x = 4

On substituting x = 4 in (i), we get:
16 + 3y = 7
⇒ 3y = (7 − 16) = −9
⇒ y = −3

Hence, the solution is x = 4 and y = −3.

Answer:

The given system of equations is
$0.4 x + 0.3 y = 1.7 . . . . . (i) 0.7 x - 0.2 y = 0.8 . . . . . (i i)$
Multiplying (i) by 0.2 and (ii) by 0.3 and adding them, we get
$0.8 x + 2.1 x = 3.4 + 2.4 \Rightarrow 2.9 x = 5.8 \Rightarrow x = \frac{5.8}{2.9} = 2$
Now, substituting x = 2 in (i), we have
$0.4 \times 2 + 0.3 y = 1.7 \Rightarrow 0.3 y = 1.7 - 0.8 \Rightarrow y = \frac{0.9}{0.3} = 3$
Hence, x = 2 and y = 3.

Question 14:

The given system of equations is
$0.4 x + 0.3 y = 1.7 . . . . . (i) 0.7 x - 0.2 y = 0.8 . . . . . (i i)$
Multiplying (i) by 0.2 and (ii) by 0.3 and adding them, we get
$0.8 x + 2.1 x = 3.4 + 2.4 \Rightarrow 2.9 x = 5.8 \Rightarrow x = \frac{5.8}{2.9} = 2$
Now, substituting x = 2 in (i), we have
$0.4 \times 2 + 0.3 y = 1.7 \Rightarrow 0.3 y = 1.7 - 0.8 \Rightarrow y = \frac{0.9}{0.3} = 3$
Hence, x = 2 and y = 3.

Answer:

The given system of equations is
$0.3 x + 0.5 y = 0.5 . . . . . (i) 0.5 x + 0.7 y = 0.74 . . . . . (i i)$
Multiplying (i) by 5 and (ii) by 3 and subtracting (ii) from (i), we get
$2.5 y - 2.1 y = 2.5 - 2.22 \Rightarrow 0.4 y = 0.28 \Rightarrow y = \frac{0.28}{0.4} = 0.7$
Now, substituting y = 0.7 in (i), we have
$0.3 x + 0.5 \times 0.7 = 0.5 \Rightarrow 0.3 x = 0.50 - 0.35 = 0.15 \Rightarrow x = \frac{0.15}{0.3} = 0.5$
Hence, x = 0.5 and y = 0.7.

Question 15:

The given system of equations is
$0.3 x + 0.5 y = 0.5 . . . . . (i) 0.5 x + 0.7 y = 0.74 . . . . . (i i)$
Multiplying (i) by 5 and (ii) by 3 and subtracting (ii) from (i), we get
$2.5 y - 2.1 y = 2.5 - 2.22 \Rightarrow 0.4 y = 0.28 \Rightarrow y = \frac{0.28}{0.4} = 0.7$
Now, substituting y = 0.7 in (i), we have
$0.3 x + 0.5 \times 0.7 = 0.5 \Rightarrow 0.3 x = 0.50 - 0.35 = 0.15 \Rightarrow x = \frac{0.15}{0.3} = 0.5$
Hence, x = 0.5 and y = 0.7.

Answer:

The given equations are:
7(y + 3) − 2(x + 2) = 14
⇒ 7y + 21 − 2x − 4 = 14
⇒ −2x + 7y = −3 ..........(i)

and 4(y − 2) + 3(x − 3) = 2
⇒ 4y − 8 + 3x − 9 = 2
⇒ 3x + 4y = 19 .........(ii)

On multiplying (i) by 4 and (ii) by 7, we get:
−8x + 28y = −12 ........(iii)
21x + 28y = 133 ...........(iv)

On subtracting (iii) from (iv), we get:
29x = 145
⇒ x = 5

On substituting x = 5 in (i), we get:
−10 + 7y = −3
⇒ 7y = (−3 + 10) = 7
⇒ y = 1

Hence, the solution is x = 5 and y = 1.

Question 16:

The given equations are:
7(y + 3) − 2(x + 2) = 14
⇒ 7y + 21 − 2x − 4 = 14
⇒ −2x + 7y = −3 ..........(i)

and 4(y − 2) + 3(x − 3) = 2
⇒ 4y − 8 + 3x − 9 = 2
⇒ 3x + 4y = 19 .........(ii)

On multiplying (i) by 4 and (ii) by 7, we get:
−8x + 28y = −12 ........(iii)
21x + 28y = 133 ...........(iv)

On subtracting (iii) from (iv), we get:
29x = 145
⇒ x = 5

On substituting x = 5 in (i), we get:
−10 + 7y = −3
⇒ 7y = (−3 + 10) = 7
⇒ y = 1

Hence, the solution is x = 5 and y = 1.

Answer:

The given equations are:
6x + 5y = 7x + 3y + 1 = 2(x + 6y − 1)
⇒ 6x + 5y = 2(x + 6y − 1)
⇒ 6x + 5y = 2x + 12y − 2
⇒ 6x − 2x + 5y − 12y = − 2
⇒ 4x − 7y = −2 ..........(i)

and 7x + 3y + 1 = 2(x + 6y − 1)
⇒ 7x + 3y + 1 = 2x + 12y − 2
⇒ 7x − 2x + 3y − 12y = −2 − 1
⇒ 5x − 9y = −3 .........(ii)

On multiplying (i) by 9 and (ii) by 7, we get:
36x − 63y = −18 ..........(iii)
35x − 63y = −21.........(iv)

On subtracting (iv) from (iii), we get:
x = (−18 + 21) = 3

On substituting x = 3 in (i), we get:
12 − 7y = −2
⇒ 7y = (2 + 12) = 14
⇒ y = 2

Hence, the solution is x = 3 and y = 2.

Question 17:

The given equations are:
6x + 5y = 7x + 3y + 1 = 2(x + 6y − 1)
⇒ 6x + 5y = 2(x + 6y − 1)
⇒ 6x + 5y = 2x + 12y − 2
⇒ 6x − 2x + 5y − 12y = − 2
⇒ 4x − 7y = −2 ..........(i)

and 7x + 3y + 1 = 2(x + 6y − 1)
⇒ 7x + 3y + 1 = 2x + 12y − 2
⇒ 7x − 2x + 3y − 12y = −2 − 1
⇒ 5x − 9y = −3 .........(ii)

On multiplying (i) by 9 and (ii) by 7, we get:
36x − 63y = −18 ..........(iii)
35x − 63y = −21.........(iv)

On subtracting (iv) from (iii), we get:
x = (−18 + 21) = 3

On substituting x = 3 in (i), we get:
12 − 7y = −2
⇒ 7y = (2 + 12) = 14
⇒ y = 2

Hence, the solution is x = 3 and y = 2.

Answer:

The given equations are:
$\frac{x + y - 8}{2} = \frac{x + 2 y - 14}{3} = \frac{3 x + y - 12}{11}$
i.e., $\frac{x + y - 8}{2} = \frac{3 x + y - 12}{11}$
By cross multiplication, we get:
11x + 11y − 88 = 6x + 2y − 24
⇒ 11x − 6x + 11y − 2y = −24 + 88
⇒ 5x + 9y = 64 .........(i)

and $\frac{x + 2 y - 14}{3} = \frac{3 x + y - 12}{11}$
⇒ 11x + 22y − 154 = 9x + 3y − 36
⇒ 11x − 9x + 22y − 3y = −36 + 154
⇒ 2x + 19y = 118 .........(ii)

On multiplying (i) by 19 and (ii) by 9, we get:
95x + 171y = 1216..........(iii)
18x + 171y = 1062.........(iv)

On subtracting (iv) from (iii), we get:
77x = 154
⇒ x = 2

On substituting x = 2 in (i), we get:
10 + 9y = 64
⇒ 9y = (64 − 10) = 54
⇒ y = 6

Hence, the solution is x = 2 and y = 6.

Question 18:

The given equations are:
$\frac{x + y - 8}{2} = \frac{x + 2 y - 14}{3} = \frac{3 x + y - 12}{11}$
i.e., $\frac{x + y - 8}{2} = \frac{3 x + y - 12}{11}$
By cross multiplication, we get:
11x + 11y − 88 = 6x + 2y − 24
⇒ 11x − 6x + 11y − 2y = −24 + 88
⇒ 5x + 9y = 64 .........(i)

and $\frac{x + 2 y - 14}{3} = \frac{3 x + y - 12}{11}$
⇒ 11x + 22y − 154 = 9x + 3y − 36
⇒ 11x − 9x + 22y − 3y = −36 + 154
⇒ 2x + 19y = 118 .........(ii)

On multiplying (i) by 19 and (ii) by 9, we get:
95x + 171y = 1216..........(iii)
18x + 171y = 1062.........(iv)

On subtracting (iv) from (iii), we get:
77x = 154
⇒ x = 2

On substituting x = 2 in (i), we get:
10 + 9y = 64
⇒ 9y = (64 − 10) = 54
⇒ y = 6

Hence, the solution is x = 2 and y = 6.

Answer:

The given equations are:
$\frac{5}{x} + 6 y = 13$ ............(i)
$\frac{3}{x} + 4 y = 7$ .............(ii)

Putting $\frac{1}{x} = u$ , we get:
5u + 6y = 13 .............(iii)
3u + 4y = 7 ...........(iv)

On multiplying (iii) by 4 and (iv) by 6, we get:
20u + 24y = 52 ...........(v)
18u + 24y = 42 ............(vi)

On subtracting (vi) from (v), we get:
2u = 10 ⇒ u = 5
$\Rightarrow \frac{1}{x} = 5 \Rightarrow x = \frac{1}{5}$
On substituting $x = \frac{1}{5}$ in (i), we get:
$\frac{5}{\frac{1}{5}} + 6 y = 13$
⇒ 25 + 6y = 13
⇒ 6y = (13 − 25) = −12
⇒ y = −2

Hence, the required solution is $x = \frac{1}{5}$ and y = −2.

Question 19:

The given equations are:
$\frac{5}{x} + 6 y = 13$ ............(i)
$\frac{3}{x} + 4 y = 7$ .............(ii)

Putting $\frac{1}{x} = u$ , we get:
5u + 6y = 13 .............(iii)
3u + 4y = 7 ...........(iv)

On multiplying (iii) by 4 and (iv) by 6, we get:
20u + 24y = 52 ...........(v)
18u + 24y = 42 ............(vi)

On subtracting (vi) from (v), we get:
2u = 10 ⇒ u = 5
$\Rightarrow \frac{1}{x} = 5 \Rightarrow x = \frac{1}{5}$
On substituting $x = \frac{1}{5}$ in (i), we get:
$\frac{5}{\frac{1}{5}} + 6 y = 13$
⇒ 25 + 6y = 13
⇒ 6y = (13 − 25) = −12
⇒ y = −2

Hence, the required solution is $x = \frac{1}{5}$ and y = −2.

Answer:

The given equations are:
$x + \frac{6}{y} = 6$ ............(i)
$3 x - \frac{8}{y} = 5$ .............(ii)
Putting $\frac{1}{y} = v$ , we get:
x + 6v = 6 .............(iii)
3x − 8v = 5 ...........(iv)

On multiplying (iii) by 4 and (iv) by 3, we get:
4x + 24v = 24 ...........(v)
9x − 24v = 15 ............(vi)

On adding (v) and (vi), we get:
13x = 39 ⇒ x = 3

On substituting x = 3 in (i), we get:
$3 + \frac{6}{y} = 6$
$\Rightarrow \frac{6}{y} = (6 - 3) = 3 \Rightarrow 3 y = 6 \Rightarrow y = 2$
Hence, the required solution is x = 3 and y = 2.

Question 20:

The given equations are:
$x + \frac{6}{y} = 6$ ............(i)
$3 x - \frac{8}{y} = 5$ .............(ii)
Putting $\frac{1}{y} = v$ , we get:
x + 6v = 6 .............(iii)
3x − 8v = 5 ...........(iv)

On multiplying (iii) by 4 and (iv) by 3, we get:
4x + 24v = 24 ...........(v)
9x − 24v = 15 ............(vi)

On adding (v) and (vi), we get:
13x = 39 ⇒ x = 3

On substituting x = 3 in (i), we get:
$3 + \frac{6}{y} = 6$
$\Rightarrow \frac{6}{y} = (6 - 3) = 3 \Rightarrow 3 y = 6 \Rightarrow y = 2$
Hence, the required solution is x = 3 and y = 2.

Answer:

The given equations are:
$2 x - \frac{3}{y} = 9$ ............(i)
$3 x + \frac{7}{y} = 2$ .............(ii)
Putting $\frac{1}{y} = v$ , we get:
2x − 3v = 9 .............(iii)
3x + 7v = 2 ...........(iv)

On multiplying (iii) by 7 and (iv) by 3, we get:
14x − 21v = 63 .............(v)
9x + 21v = 6............(vi)

On adding (v) and (vi), we get:
23x = 69 ⇒ x = 3

On substituting x = 3 in (i), we get:
$2 \times 3 - \frac{3}{y} = 9$
$\Rightarrow 6 - \frac{3}{y} = 9 \Rightarrow \frac{3}{y} = - 3 \Rightarrow y = - 1$
Hence, the required solution is x = 3 and y = −1.

Question 21:

The given equations are:
$2 x - \frac{3}{y} = 9$ ............(i)
$3 x + \frac{7}{y} = 2$ .............(ii)
Putting $\frac{1}{y} = v$ , we get:
2x − 3v = 9 .............(iii)
3x + 7v = 2 ...........(iv)

On multiplying (iii) by 7 and (iv) by 3, we get:
14x − 21v = 63 .............(v)
9x + 21v = 6............(vi)

On adding (v) and (vi), we get:
23x = 69 ⇒ x = 3

On substituting x = 3 in (i), we get:
$2 \times 3 - \frac{3}{y} = 9$
$\Rightarrow 6 - \frac{3}{y} = 9 \Rightarrow \frac{3}{y} = - 3 \Rightarrow y = - 1$
Hence, the required solution is x = 3 and y = −1.

Answer:

$Given : \frac{3}{x} + \frac{8}{y} = - 1 \frac{1}{x} - \frac{2}{y} = 2 Let \frac{1}{x} = u and \frac{1}{y} = v . . . (1) Therefore, the equations become 3 u + 8 v = - 1 . . . (2) u - 2 v = 2 . . . (3) u - 2 v = 2 \Rightarrow u = 2 + 2 v . . . (4) Substituting the value of u in (2), we get 3 (2 + 2 v) + 8 v = - 1 \Rightarrow 6 + 6 v + 8 v = - 1 \Rightarrow 14 v = - 1 - 6 \Rightarrow 14 v = - 7 \Rightarrow v = \frac{- 7}{14} \Rightarrow v = - \frac{1}{2} . . . (5) \Rightarrow y = \frac{1}{v} = \frac{1}{- \frac{1}{2}} = - 2 (∵ \frac{1}{y} = v) Substituting the value of v in (4), we get u = 2 + 2 (- \frac{1}{2}) \Rightarrow u = 2 - 1 \Rightarrow u = 1 \Rightarrow x = \frac{1}{u} = \frac{1}{1} = 1 (∵ \frac{1}{x} = u) Hence, x = 1 and y = - 2 .$

Question 22:

$Given : \frac{3}{x} + \frac{8}{y} = - 1 \frac{1}{x} - \frac{2}{y} = 2 Let \frac{1}{x} = u and \frac{1}{y} = v . . . (1) Therefore, the equations become 3 u + 8 v = - 1 . . . (2) u - 2 v = 2 . . . (3) u - 2 v = 2 \Rightarrow u = 2 + 2 v . . . (4) Substituting the value of u in (2), we get 3 (2 + 2 v) + 8 v = - 1 \Rightarrow 6 + 6 v + 8 v = - 1 \Rightarrow 14 v = - 1 - 6 \Rightarrow 14 v = - 7 \Rightarrow v = \frac{- 7}{14} \Rightarrow v = - \frac{1}{2} . . . (5) \Rightarrow y = \frac{1}{v} = \frac{1}{- \frac{1}{2}} = - 2 (∵ \frac{1}{y} = v) Substituting the value of v in (4), we get u = 2 + 2 (- \frac{1}{2}) \Rightarrow u = 2 - 1 \Rightarrow u = 1 \Rightarrow x = \frac{1}{u} = \frac{1}{1} = 1 (∵ \frac{1}{x} = u) Hence, x = 1 and y = - 2 .$

Answer:

The given equations are:
$\frac{9}{x} - \frac{4}{y} = 8$ ............(i)
$\frac{13}{x} + \frac{7}{y} = 101$ .............(ii)

Putting $\frac{1}{x} = u$ and $\frac{1}{y} = v$ , we get:
9u − 4v = 8 .............(iii)
13u + 7v = 101 ...........(iv)

On multiplying (iii) by 7 and (iv) by 4, we get:
63u − 28v = 56 .............(v)
52u + 28v = 404............(vi)

On adding (v) and (vi), we get:
115u = 460 ⇒ u = 4
$\Rightarrow \frac{1}{x} = 4 \Rightarrow x = \frac{1}{4}$
On substituting $x = \frac{1}{4}$ in (i), we get:
$\frac{9}{\frac{1}{4}} - \frac{4}{y} = 8$
$\Rightarrow 36 - \frac{4}{y} = 8 \Rightarrow \frac{4}{y} = (36 - 8) = 28$
$\Rightarrow y = \frac{4}{28} = \frac{1}{7}$
Hence, the required solution is $x = \frac{1}{4}$ and $y = \frac{1}{7}$ .

Question 23:

The given equations are:
$\frac{9}{x} - \frac{4}{y} = 8$ ............(i)
$\frac{13}{x} + \frac{7}{y} = 101$ .............(ii)

Putting $\frac{1}{x} = u$ and $\frac{1}{y} = v$ , we get:
9u − 4v = 8 .............(iii)
13u + 7v = 101 ...........(iv)

On multiplying (iii) by 7 and (iv) by 4, we get:
63u − 28v = 56 .............(v)
52u + 28v = 404............(vi)

On adding (v) and (vi), we get:
115u = 460 ⇒ u = 4
$\Rightarrow \frac{1}{x} = 4 \Rightarrow x = \frac{1}{4}$
On substituting $x = \frac{1}{4}$ in (i), we get:
$\frac{9}{\frac{1}{4}} - \frac{4}{y} = 8$
$\Rightarrow 36 - \frac{4}{y} = 8 \Rightarrow \frac{4}{y} = (36 - 8) = 28$
$\Rightarrow y = \frac{4}{28} = \frac{1}{7}$
Hence, the required solution is $x = \frac{1}{4}$ and $y = \frac{1}{7}$ .

Answer:

The given equations are:
$\frac{5}{x} - \frac{3}{y} = 1$ ............(i)
$\frac{3}{2 x} + \frac{2}{3 y} = 5$ .............(ii)

Putting $\frac{1}{x} = u$ and $\frac{1}{y} = v$ , we get:
5u − 3v = 1 .............(iii)
⇒ $\frac{3}{2} u + \frac{2}{3} v = 5$
$\Rightarrow \frac{9 u + 4 v}{6} = 5$
$\Rightarrow 9 u + 4 v = 30$ ...............(iv)

On multiplying (iii) by 4 and (iv) by 3, we get:
20u − 12v = 4 .............(v)
27u + 12v = 90 ...........(vi)

On adding (iv) and (v), we get:
47u = 94 ⇒ u = 2
$\Rightarrow \frac{1}{x} = 2 \Rightarrow x = \frac{1}{2}$

On substituting $x = \frac{1}{2}$ in (i), we get:
$\frac{5}{\frac{1}{2}} - \frac{3}{y} = 1$
$\Rightarrow 10 - \frac{3}{y} = 1 \Rightarrow \frac{3}{y} = (10 - 1) = 9$
$\Rightarrow y = \frac{3}{9} = \frac{1}{3}$
Hence, the required solution is $x = \frac{1}{2}$ and $y = \frac{1}{3}$ .

Question 24:

The given equations are:
$\frac{5}{x} - \frac{3}{y} = 1$ ............(i)
$\frac{3}{2 x} + \frac{2}{3 y} = 5$ .............(ii)

Putting $\frac{1}{x} = u$ and $\frac{1}{y} = v$ , we get:
5u − 3v = 1 .............(iii)
⇒ $\frac{3}{2} u + \frac{2}{3} v = 5$
$\Rightarrow \frac{9 u + 4 v}{6} = 5$
$\Rightarrow 9 u + 4 v = 30$ ...............(iv)

On multiplying (iii) by 4 and (iv) by 3, we get:
20u − 12v = 4 .............(v)
27u + 12v = 90 ...........(vi)

On adding (iv) and (v), we get:
47u = 94 ⇒ u = 2
$\Rightarrow \frac{1}{x} = 2 \Rightarrow x = \frac{1}{2}$

On substituting $x = \frac{1}{2}$ in (i), we get:
$\frac{5}{\frac{1}{2}} - \frac{3}{y} = 1$
$\Rightarrow 10 - \frac{3}{y} = 1 \Rightarrow \frac{3}{y} = (10 - 1) = 9$
$\Rightarrow y = \frac{3}{9} = \frac{1}{3}$
Hence, the required solution is $x = \frac{1}{2}$ and $y = \frac{1}{3}$ .

Answer:

Multiplying equation (i) and (ii) by 6, we get
$\frac{3}{x} + \frac{2}{y} = 12 . . . . . (i) \frac{2}{x} + \frac{3}{y} = 13 . . . . . (i i)$
Multiplying (i) by 3 and (ii) by 2 and subtracting (ii) from (i), we get
$\frac{9}{x} - \frac{4}{x} = 36 - 26 \Rightarrow \frac{5}{x} = 10 \Rightarrow x = \frac{5}{10} = \frac{1}{2}$
Now, substituting $x = \frac{1}{2}$ in (i), we have
$6 + \frac{2}{y} = 12 \Rightarrow \frac{2}{y} = 6 \Rightarrow y = \frac{1}{3}$
Hence, $x = \frac{1}{2}$ and $y = \frac{1}{3}$ .

Question 25:

Multiplying equation (i) and (ii) by 6, we get
$\frac{3}{x} + \frac{2}{y} = 12 . . . . . (i) \frac{2}{x} + \frac{3}{y} = 13 . . . . . (i i)$
Multiplying (i) by 3 and (ii) by 2 and subtracting (ii) from (i), we get
$\frac{9}{x} - \frac{4}{x} = 36 - 26 \Rightarrow \frac{5}{x} = 10 \Rightarrow x = \frac{5}{10} = \frac{1}{2}$
Now, substituting $x = \frac{1}{2}$ in (i), we have
$6 + \frac{2}{y} = 12 \Rightarrow \frac{2}{y} = 6 \Rightarrow y = \frac{1}{3}$
Hence, $x = \frac{1}{2}$ and $y = \frac{1}{3}$ .

Answer:

The given equations are:
4x + 6y = 3xy .......(i)
8x + 9y = 5xy .........(ii)

From equation (i), we have:

$\frac{4 x + 6 y}{x y} = 3$
$\Rightarrow \frac{4}{y} + \frac{6}{x} = 3$ .............(iii)

For equation (ii), we have:

$\frac{8 x + 9 y}{x y} = 5$
$\Rightarrow \frac{8}{y} + \frac{9}{x} = 5$ .............(iv)
On substituting $\frac{1}{y} = v and \frac{1}{x} = u$ , we get:
4v + 6u = 3 ...........(v)
8v + 9u = 5 ...........(vi)

On multiplying (v) by 9 and (vi) by 6, we get:
36v + 54u = 27 ..............(vii)
48v + 54u = 30 ...............(viii)

On subtracting (vii) from (viii), we get:
12v = 3 $\Rightarrow v = \frac{3}{12} = \frac{1}{4}$
$\Rightarrow \frac{1}{y} = \frac{1}{4} \Rightarrow y = 4$

On substituting y = 4 in (iii), we get:
$\frac{4}{4} + \frac{6}{x} = 3$
$\Rightarrow 1 + \frac{6}{x} = 3 \Rightarrow \frac{6}{x} = (3 - 1) = 2$
$\Rightarrow 2 x = 6 \Rightarrow x = \frac{6}{2} = 3$
Hence, the required solution is x = 3 and y = 4.

Question 26:

The given equations are:
4x + 6y = 3xy .......(i)
8x + 9y = 5xy .........(ii)

From equation (i), we have:

$\frac{4 x + 6 y}{x y} = 3$
$\Rightarrow \frac{4}{y} + \frac{6}{x} = 3$ .............(iii)

For equation (ii), we have:

$\frac{8 x + 9 y}{x y} = 5$
$\Rightarrow \frac{8}{y} + \frac{9}{x} = 5$ .............(iv)
On substituting $\frac{1}{y} = v and \frac{1}{x} = u$ , we get:
4v + 6u = 3 ...........(v)
8v + 9u = 5 ...........(vi)

On multiplying (v) by 9 and (vi) by 6, we get:
36v + 54u = 27 ..............(vii)
48v + 54u = 30 ...............(viii)

On subtracting (vii) from (viii), we get:
12v = 3 $\Rightarrow v = \frac{3}{12} = \frac{1}{4}$
$\Rightarrow \frac{1}{y} = \frac{1}{4} \Rightarrow y = 4$

On substituting y = 4 in (iii), we get:
$\frac{4}{4} + \frac{6}{x} = 3$
$\Rightarrow 1 + \frac{6}{x} = 3 \Rightarrow \frac{6}{x} = (3 - 1) = 2$
$\Rightarrow 2 x = 6 \Rightarrow x = \frac{6}{2} = 3$
Hence, the required solution is x = 3 and y = 4.

Answer:

The given equations are:
x + y = 5xy .......(i)
3x + 2y = 13xy .........(ii)

From equation (i), we have:

$\frac{x + y}{x y} = 5$
$\Rightarrow \frac{1}{y} + \frac{1}{x} = 5$ .............(iii)

From equation (ii), we have:

$\frac{3 x + 2 y}{x y} = 13$
$\Rightarrow \frac{3}{y} + \frac{2}{x} = 13$ .............(iv)
On substituting $\frac{1}{y} = v and \frac{1}{x} = u$ , we get:
v + u = 5 ...........(v)
3v + 2u = 13 ...........(vi)

On multiplying (v) by 2, we get:
2v + 2u = 10 ..............(vii)
On subtracting (vii) from (vi), we get:
v = 3
$\Rightarrow \frac{1}{y} = 3 \Rightarrow y = \frac{1}{3}$
On substituting $y = \frac{1}{3}$ in (iii), we get:
$\frac{1}{\frac{1}{3}} + \frac{1}{x} = 5$
$\Rightarrow 3 + \frac{1}{x} = 5 \Rightarrow \frac{1}{x} = 2 \Rightarrow x = \frac{1}{2}$
Hence, the required solution is $x = \frac{1}{2}$ and $y = \frac{1}{3}$ or x = 0 and y = 0.

Question 27:

The given equations are:
x + y = 5xy .......(i)
3x + 2y = 13xy .........(ii)

From equation (i), we have:

$\frac{x + y}{x y} = 5$
$\Rightarrow \frac{1}{y} + \frac{1}{x} = 5$ .............(iii)

From equation (ii), we have:

$\frac{3 x + 2 y}{x y} = 13$
$\Rightarrow \frac{3}{y} + \frac{2}{x} = 13$ .............(iv)
On substituting $\frac{1}{y} = v and \frac{1}{x} = u$ , we get:
v + u = 5 ...........(v)
3v + 2u = 13 ...........(vi)

On multiplying (v) by 2, we get:
2v + 2u = 10 ..............(vii)
On subtracting (vii) from (vi), we get:
v = 3
$\Rightarrow \frac{1}{y} = 3 \Rightarrow y = \frac{1}{3}$
On substituting $y = \frac{1}{3}$ in (iii), we get:
$\frac{1}{\frac{1}{3}} + \frac{1}{x} = 5$
$\Rightarrow 3 + \frac{1}{x} = 5 \Rightarrow \frac{1}{x} = 2 \Rightarrow x = \frac{1}{2}$
Hence, the required solution is $x = \frac{1}{2}$ and $y = \frac{1}{3}$ or x = 0 and y = 0.

Answer:

The given equations are
$\frac{5}{x + y} - \frac{2}{x - y} = - 1 . . . . . (i) \frac{15}{x + y} + \frac{7}{x - y} = 10 . . . . . (i i)$
Substituting $\frac{1}{x + y} = u and \frac{1}{x - y} = v$ in (i) and (ii), we get
$5 u - 2 v = - 1 . . . . . (i i i) 15 u + 7 v = 10 . . . . . (i v)$
Multiplying (iii) by 3 and subtracting it from (iv), we get
$7 v + 6 v = 10 + 3 \Rightarrow 13 v = 13 \Rightarrow v = 1 \Rightarrow x - y = 1 (∵ \frac{1}{x - y} = v) . . . . . (v)$
Now, substituting v = 1 in (iii), we get
$5 u - 2 = - 1 \Rightarrow 5 u = 1 \Rightarrow u = \frac{1}{5} x + y = 5 . . . . . (v i)$
Adding (v) and (vi), we get
$2 x = 6 \Rightarrow x = 3$
Substituting x = 3 in (vi), we have
$3 + y = 5 \Rightarrow y = 5 - 3 = 2$
Hence, x = 3 and y = 2.

Question 28:

The given equations are
$\frac{5}{x + y} - \frac{2}{x - y} = - 1 . . . . . (i) \frac{15}{x + y} + \frac{7}{x - y} = 10 . . . . . (i i)$
Substituting $\frac{1}{x + y} = u and \frac{1}{x - y} = v$ in (i) and (ii), we get
$5 u - 2 v = - 1 . . . . . (i i i) 15 u + 7 v = 10 . . . . . (i v)$
Multiplying (iii) by 3 and subtracting it from (iv), we get
$7 v + 6 v = 10 + 3 \Rightarrow 13 v = 13 \Rightarrow v = 1 \Rightarrow x - y = 1 (∵ \frac{1}{x - y} = v) . . . . . (v)$
Now, substituting v = 1 in (iii), we get
$5 u - 2 = - 1 \Rightarrow 5 u = 1 \Rightarrow u = \frac{1}{5} x + y = 5 . . . . . (v i)$
Adding (v) and (vi), we get
$2 x = 6 \Rightarrow x = 3$
Substituting x = 3 in (vi), we have
$3 + y = 5 \Rightarrow y = 5 - 3 = 2$
Hence, x = 3 and y = 2.

Answer:

The given equations are:
$\frac{3}{x + y} + \frac{2}{x - y} = 2$           ...(i)
$\frac{9}{x + y} - \frac{4}{x - y} = 1$            ...(ii)
Putting $\frac{1}{x + y} = u$ and $\frac{1}{x - y} = v$ , we get:
3u + 2v = 2            ...(iii)
9u − 4v = 1              ...(iv)
On multiplying (iii) by 2, we get:
6u + 4v = 4               ...(v)
On adding (iv) and (v), we get:
15u = 5
$\Rightarrow u = \frac{5}{15} = \frac{1}{3}$
$\Rightarrow \frac{1}{x + y} = \frac{1}{3} \Rightarrow x + y = 3$              ...(vi)
On substituting $u = \frac{1}{3}$ in (iii), we get:
1 + 2v = 2
⇒ 2v = 1
$\Rightarrow v = \frac{1}{2}$
$\Rightarrow \frac{1}{x - y} = \frac{1}{2} \Rightarrow x - y = 2$                 ...(vii)
On adding (vi) and (vii), we get:
2x = 5
⇒ $x = \frac{5}{2}$
On substituting $x = \frac{5}{2}$ in (vi), we get:
$\frac{5}{2} + y = 3 \Rightarrow y = (3 - \frac{5}{2}) = \frac{1}{2}$
Hence, the required solution is $x = \frac{5}{2} and y = \frac{1}{2}$ .

Question 29:

The given equations are:
$\frac{3}{x + y} + \frac{2}{x - y} = 2$           ...(i)
$\frac{9}{x + y} - \frac{4}{x - y} = 1$            ...(ii)
Putting $\frac{1}{x + y} = u$ and $\frac{1}{x - y} = v$ , we get:
3u + 2v = 2            ...(iii)
9u − 4v = 1              ...(iv)
On multiplying (iii) by 2, we get:
6u + 4v = 4               ...(v)
On adding (iv) and (v), we get:
15u = 5
$\Rightarrow u = \frac{5}{15} = \frac{1}{3}$
$\Rightarrow \frac{1}{x + y} = \frac{1}{3} \Rightarrow x + y = 3$              ...(vi)
On substituting $u = \frac{1}{3}$ in (iii), we get:
1 + 2v = 2
⇒ 2v = 1
$\Rightarrow v = \frac{1}{2}$
$\Rightarrow \frac{1}{x - y} = \frac{1}{2} \Rightarrow x - y = 2$                 ...(vii)
On adding (vi) and (vii), we get:
2x = 5
⇒ $x = \frac{5}{2}$
On substituting $x = \frac{5}{2}$ in (vi), we get:
$\frac{5}{2} + y = 3 \Rightarrow y = (3 - \frac{5}{2}) = \frac{1}{2}$
Hence, the required solution is $x = \frac{5}{2} and y = \frac{1}{2}$ .

Answer:

The given equations are:
$\frac{5}{x + 1} - \frac{2}{y - 1} = \frac{1}{2}$ .............(i)
$\frac{10}{x + 1} + \frac{2}{y - 1} = \frac{5}{2}$ ..............(ii)
Putting $\frac{1}{x + 1} = u$ and $\frac{1}{y - 1} = v$ , we get:
$5 u - 2 v = \frac{1}{2}$ .................(iii)
$10 u + 2 v = \frac{5}{2}$ ................(iv)
On adding (iii) and (iv), we get:
15u = 3
⇒ $u = \frac{3}{15} = \frac{1}{5}$
⇒ $\frac{1}{x + 1} = \frac{1}{5} \Rightarrow x + 1 = 5 \Rightarrow x = 4$
On substituting $u = \frac{1}{5}$ in (iii), we get:
$5 \times \frac{1}{5} - 2 v = \frac{1}{2} \Rightarrow 1 - 2 v = \frac{1}{2}$
$\Rightarrow 2 v = \frac{1}{2} \Rightarrow v = \frac{1}{4}$
$\Rightarrow \frac{1}{y - 1} = \frac{1}{4} \Rightarrow y - 1 = 4 \Rightarrow y = 5$
Hence, the required solution is x = 4 and y = 5.

Question 30:

The given equations are:
$\frac{5}{x + 1} - \frac{2}{y - 1} = \frac{1}{2}$ .............(i)
$\frac{10}{x + 1} + \frac{2}{y - 1} = \frac{5}{2}$ ..............(ii)
Putting $\frac{1}{x + 1} = u$ and $\frac{1}{y - 1} = v$ , we get:
$5 u - 2 v = \frac{1}{2}$ .................(iii)
$10 u + 2 v = \frac{5}{2}$ ................(iv)
On adding (iii) and (iv), we get:
15u = 3
⇒ $u = \frac{3}{15} = \frac{1}{5}$
⇒ $\frac{1}{x + 1} = \frac{1}{5} \Rightarrow x + 1 = 5 \Rightarrow x = 4$
On substituting $u = \frac{1}{5}$ in (iii), we get:
$5 \times \frac{1}{5} - 2 v = \frac{1}{2} \Rightarrow 1 - 2 v = \frac{1}{2}$
$\Rightarrow 2 v = \frac{1}{2} \Rightarrow v = \frac{1}{4}$
$\Rightarrow \frac{1}{y - 1} = \frac{1}{4} \Rightarrow y - 1 = 4 \Rightarrow y = 5$
Hence, the required solution is x = 4 and y = 5.

Answer:

The given equations are:
$\frac{44}{x + y} + \frac{30}{x - y} = 10$         ...(i)
$\frac{55}{x + y} + \frac{40}{x - y} = 13$          ...(ii)
Putting $\frac{1}{x + y} = u$ and $\frac{1}{x - y} = v$ , we get:
44u + 30v = 10          ...(iii)
55u + 40v = 13          ...(iv)
On multiplying (iii) by 4 and (iv) by 3, we get:
176u + 120v = 40           ...(v)
165u + 120v = 39           ...(vi)
On subtracting (vi) from (v), we get:
11u = 1
$\Rightarrow u = \frac{1}{11}$
$\Rightarrow \frac{1}{x + y} = \frac{1}{11} \Rightarrow x + y = 11$          ...(vii)
On substituting $u = \frac{1}{11}$ in (iii), we get:
4 + 30v = 10
⇒ 30v = 6
$\Rightarrow v = \frac{6}{30} = \frac{1}{5}$
$\Rightarrow \frac{1}{x - y} = \frac{1}{5} \Rightarrow x - y = 5$      ...(viii)
On adding (vii) and (viii), we get:
2x = 16
⇒ x = 8
On substituting x = 8 in (vii), we get:
8 + y = 11
⇒ y = (11 − 8) = 3
Hence, the required solution is x = 8 and y = 3.

Question 31:

The given equations are:
$\frac{44}{x + y} + \frac{30}{x - y} = 10$         ...(i)
$\frac{55}{x + y} + \frac{40}{x - y} = 13$          ...(ii)
Putting $\frac{1}{x + y} = u$ and $\frac{1}{x - y} = v$ , we get:
44u + 30v = 10          ...(iii)
55u + 40v = 13          ...(iv)
On multiplying (iii) by 4 and (iv) by 3, we get:
176u + 120v = 40           ...(v)
165u + 120v = 39           ...(vi)
On subtracting (vi) from (v), we get:
11u = 1
$\Rightarrow u = \frac{1}{11}$
$\Rightarrow \frac{1}{x + y} = \frac{1}{11} \Rightarrow x + y = 11$          ...(vii)
On substituting $u = \frac{1}{11}$ in (iii), we get:
4 + 30v = 10
⇒ 30v = 6
$\Rightarrow v = \frac{6}{30} = \frac{1}{5}$
$\Rightarrow \frac{1}{x - y} = \frac{1}{5} \Rightarrow x - y = 5$      ...(viii)
On adding (vii) and (viii), we get:
2x = 16
⇒ x = 8
On substituting x = 8 in (vii), we get:
8 + y = 11
⇒ y = (11 − 8) = 3
Hence, the required solution is x = 8 and y = 3.

Answer:

The given equations are
$\frac{10}{x + y} + \frac{2}{x - y} = 4 . . . . . (i) \frac{15}{x + y} - \frac{9}{x - y} = - 2 . . . . . (i i)$
Substituting $\frac{1}{x + y} = u and \frac{1}{x - y} = v$ in (i) and (ii), we get
$10 u + 2 v = 4 . . . . . (i i i) 15 u - 9 v = - 2 . . . . . (i v)$
Multiplying (iii) by 9 and (iv) by 2 and adding, we get
$90 u + 30 u = 36 - 4 \Rightarrow 120 u = 32 \Rightarrow u = \frac{32}{120} = \frac{4}{15} \Rightarrow x + y = \frac{15}{4} (∵ \frac{1}{x + y} = u) . . . . . (v)$
Now, substituting $u = \frac{4}{15}$ in (iii), we get
$10 \times \frac{4}{15} + 2 v = 4 \frac{8}{3} + 2 v = 4 \Rightarrow 2 v = 4 - \frac{8}{3} = \frac{4}{3} \Rightarrow v = \frac{2}{3} \Rightarrow x - y = \frac{3}{2} (∵ \frac{1}{x - y} = v) . . . . . (v i)$
Adding (v) and (vi), we get
$2 x = \frac{15}{4} + \frac{3}{2} \Rightarrow 2 x = \frac{21}{4} \Rightarrow x = \frac{21}{8}$
Substituting $x = \frac{21}{8}$ in (v), we have
$\frac{21}{8} + y = \frac{15}{4} \Rightarrow y = \frac{15}{4} - \frac{21}{8} = \frac{9}{8}$
Hence, $x = \frac{21}{8} and y = \frac{9}{8}$ .

Question 32:

The given equations are
$\frac{10}{x + y} + \frac{2}{x - y} = 4 . . . . . (i) \frac{15}{x + y} - \frac{9}{x - y} = - 2 . . . . . (i i)$
Substituting $\frac{1}{x + y} = u and \frac{1}{x - y} = v$ in (i) and (ii), we get
$10 u + 2 v = 4 . . . . . (i i i) 15 u - 9 v = - 2 . . . . . (i v)$
Multiplying (iii) by 9 and (iv) by 2 and adding, we get
$90 u + 30 u = 36 - 4 \Rightarrow 120 u = 32 \Rightarrow u = \frac{32}{120} = \frac{4}{15} \Rightarrow x + y = \frac{15}{4} (∵ \frac{1}{x + y} = u) . . . . . (v)$
Now, substituting $u = \frac{4}{15}$ in (iii), we get
$10 \times \frac{4}{15} + 2 v = 4 \frac{8}{3} + 2 v = 4 \Rightarrow 2 v = 4 - \frac{8}{3} = \frac{4}{3} \Rightarrow v = \frac{2}{3} \Rightarrow x - y = \frac{3}{2} (∵ \frac{1}{x - y} = v) . . . . . (v i)$
Adding (v) and (vi), we get
$2 x = \frac{15}{4} + \frac{3}{2} \Rightarrow 2 x = \frac{21}{4} \Rightarrow x = \frac{21}{8}$
Substituting $x = \frac{21}{8}$ in (v), we have
$\frac{21}{8} + y = \frac{15}{4} \Rightarrow y = \frac{15}{4} - \frac{21}{8} = \frac{9}{8}$
Hence, $x = \frac{21}{8} and y = \frac{9}{8}$ .

Answer:

The given equations are:
71x + 37y = 253 ..............(i)
37x + 71y = 287 ............(ii)

On adding (i) and (ii), we get:
108x + 108y = 540
⇒ 108(x + y) = 540
⇒ (x + y) = 5................(iii)

On subtracting (ii) from (i), we get:
34x − 34y = −34
⇒ 34(x − y) = −34
⇒ (x − y) = −1...........(iv)

On adding (iii) and (iv), we get:
2x = 5 − 1= 4
⇒ x = 2

On subtracting (iv) from (iii), we get:
2y = 5 + 1 = 6
⇒ y = 3

Hence, the required solution is x = 2 and y = 3.

Question 33:

The given equations are:
71x + 37y = 253 ..............(i)
37x + 71y = 287 ............(ii)

On adding (i) and (ii), we get:
108x + 108y = 540
⇒ 108(x + y) = 540
⇒ (x + y) = 5................(iii)

On subtracting (ii) from (i), we get:
34x − 34y = −34
⇒ 34(x − y) = −34
⇒ (x − y) = −1...........(iv)

On adding (iii) and (iv), we get:
2x = 5 − 1= 4
⇒ x = 2

On subtracting (iv) from (iii), we get:
2y = 5 + 1 = 6
⇒ y = 3

Hence, the required solution is x = 2 and y = 3.

Answer:

The given equations are:
217x + 131y = 913 .............(i)
131x + 217y = 827 .............(ii)

On adding (i) and (ii), we get:
348x + 348y = 1740
⇒ 348(x + y) = 1740
⇒ x + y = 5 ............(iii)

On subtracting (ii) from (i), we get:
86x − 86y = 86
⇒ 86(x − y) = 86
⇒ x − y = 1 ...............(iv)

On adding (iii) and (iv), we get :
2x = 6 ⇒ x = 3

On substituting x = 3 in (iii), we get:
3 + y = 5
⇒ y = 5 − 3 = 2

Hence, the required solution is x = 3 and y = 2.

Question 34:

The given equations are:
217x + 131y = 913 .............(i)
131x + 217y = 827 .............(ii)

On adding (i) and (ii), we get:
348x + 348y = 1740
⇒ 348(x + y) = 1740
⇒ x + y = 5 ............(iii)

On subtracting (ii) from (i), we get:
86x − 86y = 86
⇒ 86(x − y) = 86
⇒ x − y = 1 ...............(iv)

On adding (iii) and (iv), we get :
2x = 6 ⇒ x = 3

On substituting x = 3 in (iii), we get:
3 + y = 5
⇒ y = 5 − 3 = 2

Hence, the required solution is x = 3 and y = 2.

Answer:

The given equations are
$23 x - 29 y = 98 . . . . . (i) 29 x - 23 y = 110 . . . . . (i i)$
Adding (i) and (ii), we get
$52 x - 52 y = 208 \Rightarrow x - y = 4 . . . . . (i i i)$
Subtracting (i) from (ii), we get
$6 x + 6 y = 12 \Rightarrow x + y = 2 . . . . . (i v)$
Now, adding equation (iii) and (iv), we get
$2 x = 6 \Rightarrow x = 3$
Substituting x = 3 in (iv), we have
$3 + y = 2 \Rightarrow y = 2 - 3 = - 1$
Hence, $x = 3 and y = - 1$ .

Question 35:

The given equations are
$23 x - 29 y = 98 . . . . . (i) 29 x - 23 y = 110 . . . . . (i i)$
Adding (i) and (ii), we get
$52 x - 52 y = 208 \Rightarrow x - y = 4 . . . . . (i i i)$
Subtracting (i) from (ii), we get
$6 x + 6 y = 12 \Rightarrow x + y = 2 . . . . . (i v)$
Now, adding equation (iii) and (iv), we get
$2 x = 6 \Rightarrow x = 3$
Substituting x = 3 in (iv), we have
$3 + y = 2 \Rightarrow y = 2 - 3 = - 1$
Hence, $x = 3 and y = - 1$ .

Answer:

The given equations can be written as
$\frac{5}{x} + \frac{2}{y} = 6 . . . . . (i) \frac{- 5}{x} + \frac{4}{y} = - 3 . . . . . (i i)$
Adding (i) and (ii), we get
$\frac{6}{y} = 3 \Rightarrow y = 2$
Substituting y = 2 in (i), we have
$\frac{5}{x} + \frac{2}{2} = 6 \Rightarrow x = 1$
Hence, x = 1 and y = 2..

Question 36:

The given equations can be written as
$\frac{5}{x} + \frac{2}{y} = 6 . . . . . (i) \frac{- 5}{x} + \frac{4}{y} = - 3 . . . . . (i i)$
Adding (i) and (ii), we get
$\frac{6}{y} = 3 \Rightarrow y = 2$
Substituting y = 2 in (i), we have
$\frac{5}{x} + \frac{2}{2} = 6 \Rightarrow x = 1$
Hence, x = 1 and y = 2..

Answer:

The given equations are
$\frac{1}{3 x + y} + \frac{1}{3 x - y} = \frac{3}{4} . . . . . (i) \frac{1}{2 (3 x + y)} - \frac{1}{2 (3 x - y)} = - \frac{1}{8} \frac{1}{3 x + y} - \frac{1}{3 x - y} = - \frac{1}{4} (Multiplying by 2) . . . . . (i i)$
Substituting $\frac{1}{3 x + y} = u and \frac{1}{3 x - y} = v$ in (i) and (ii), we get
$u + v = \frac{3}{4} . . . . . (i i i) u - v = - \frac{1}{4} . . . . . (i v)$
Adding (iii) and (iv), we get
$2 u = \frac{1}{2} \Rightarrow u = \frac{1}{4} \Rightarrow 3 x + y = 4 (∵ \frac{1}{3 x + y} = u) . . . . . (v)$
Now, substituting $u = \frac{1}{4}$ in (iii), we get
$\frac{1}{4} + v = \frac{3}{4} v = \frac{3}{4} - \frac{1}{4} \Rightarrow v = \frac{1}{2} \Rightarrow 3 x - y = 2 (∵ \frac{1}{3 x - y} = v) . . . . . (v i)$
Adding (v) and (vi), we get
$6 x = 6 \Rightarrow x = 1$
Substituting x = 1 in (v), we have
$3 + y = 4 \Rightarrow y = 1$
Hence, x = 1 and y = 1.

Question 37:

The given equations are
$\frac{1}{3 x + y} + \frac{1}{3 x - y} = \frac{3}{4} . . . . . (i) \frac{1}{2 (3 x + y)} - \frac{1}{2 (3 x - y)} = - \frac{1}{8} \frac{1}{3 x + y} - \frac{1}{3 x - y} = - \frac{1}{4} (Multiplying by 2) . . . . . (i i)$
Substituting $\frac{1}{3 x + y} = u and \frac{1}{3 x - y} = v$ in (i) and (ii), we get
$u + v = \frac{3}{4} . . . . . (i i i) u - v = - \frac{1}{4} . . . . . (i v)$
Adding (iii) and (iv), we get
$2 u = \frac{1}{2} \Rightarrow u = \frac{1}{4} \Rightarrow 3 x + y = 4 (∵ \frac{1}{3 x + y} = u) . . . . . (v)$
Now, substituting $u = \frac{1}{4}$ in (iii), we get
$\frac{1}{4} + v = \frac{3}{4} v = \frac{3}{4} - \frac{1}{4} \Rightarrow v = \frac{1}{2} \Rightarrow 3 x - y = 2 (∵ \frac{1}{3 x - y} = v) . . . . . (v i)$
Adding (v) and (vi), we get
$6 x = 6 \Rightarrow x = 1$
Substituting x = 1 in (v), we have
$3 + y = 4 \Rightarrow y = 1$
Hence, x = 1 and y = 1.

Answer:

The given equations are:
$\frac{1}{2 (x + 2 y)} + \frac{5}{3 (3 x - 2 y)} = - \frac{3}{2}$               ...(i)
$\frac{5}{4 (x + 2 y)} - \frac{3}{5 (3 x - 2 y)} = \frac{61}{60}$               ...(ii)
Putting $\frac{1}{x + 2 y} = u$ and $\frac{1}{3 x - 2 y} = v$ , we get:
$\frac{1}{2} u + \frac{5}{3} v = - \frac{3}{2}$          ...(iii)
$\frac{5}{4} u - \frac{3}{5} v = \frac{61}{60}$           ...(iv)
On multiplying (iii) by 6 and (iv) by 20, we get:
3u + 10v = −9         ...(v)
$25 u - 12 v = \frac{61}{3}$            ...(vi)
On multiplying (v) by 6 and (vi) by 5, we get:
18u + 60v = −54       ...(vii)
$125 u - 60 v = \frac{305}{3}$           ...(viii)
On adding (vii) and (viii), we get:
$143 u = \frac{305}{3} - 54 = \frac{305 - 162}{3} = \frac{143}{3}$
$\Rightarrow u = \frac{1}{3} = \frac{1}{x + 2 y}$
⇒ x + 2y = 3    ...(ix)
On substituting $u = \frac{1}{3}$ in (v), we get:
1 + 10v = −9
⇒ 10v = −10
⇒ v = −1
$\Rightarrow \frac{1}{3 x - 2 y} = - 1 \Rightarrow 3 x - 2 y = - 1$             ...(x)
On adding (ix) and (x), we get:
4x = 2
⇒ $x = \frac{1}{2}$
On substituting $x = \frac{1}{2}$ in (x), we get:
$\frac{3}{2} - 2 y = - 1 \Rightarrow 2 y = (\frac{3}{2} + 1) = \frac{5}{2} \Rightarrow y = \frac{5}{4}$
Hence, the required solution is $x = \frac{1}{2} and y = \frac{5}{4}$ .

Question 38:

The given equations are:
$\frac{1}{2 (x + 2 y)} + \frac{5}{3 (3 x - 2 y)} = - \frac{3}{2}$               ...(i)
$\frac{5}{4 (x + 2 y)} - \frac{3}{5 (3 x - 2 y)} = \frac{61}{60}$               ...(ii)
Putting $\frac{1}{x + 2 y} = u$ and $\frac{1}{3 x - 2 y} = v$ , we get:
$\frac{1}{2} u + \frac{5}{3} v = - \frac{3}{2}$          ...(iii)
$\frac{5}{4} u - \frac{3}{5} v = \frac{61}{60}$           ...(iv)
On multiplying (iii) by 6 and (iv) by 20, we get:
3u + 10v = −9         ...(v)
$25 u - 12 v = \frac{61}{3}$            ...(vi)
On multiplying (v) by 6 and (vi) by 5, we get:
18u + 60v = −54       ...(vii)
$125 u - 60 v = \frac{305}{3}$           ...(viii)
On adding (vii) and (viii), we get:
$143 u = \frac{305}{3} - 54 = \frac{305 - 162}{3} = \frac{143}{3}$
$\Rightarrow u = \frac{1}{3} = \frac{1}{x + 2 y}$
⇒ x + 2y = 3    ...(ix)
On substituting $u = \frac{1}{3}$ in (v), we get:
1 + 10v = −9
⇒ 10v = −10
⇒ v = −1
$\Rightarrow \frac{1}{3 x - 2 y} = - 1 \Rightarrow 3 x - 2 y = - 1$             ...(x)
On adding (ix) and (x), we get:
4x = 2
⇒ $x = \frac{1}{2}$
On substituting $x = \frac{1}{2}$ in (x), we get:
$\frac{3}{2} - 2 y = - 1 \Rightarrow 2 y = (\frac{3}{2} + 1) = \frac{5}{2} \Rightarrow y = \frac{5}{4}$
Hence, the required solution is $x = \frac{1}{2} and y = \frac{5}{4}$ .

Answer:

The given equations are
$\frac{2}{3 x + 2 y} + \frac{3}{3 x - 2 y} = \frac{17}{5} . . . . . (i) \frac{5}{3 x + 2 y} + \frac{1}{3 x - 2 y} = 2 . . . . . (i i)$
Substituting $\frac{1}{3 x + 2 y} = u and \frac{1}{3 x - 2 y} = v$ in (i) and (ii), we get
$2 u + 3 v = \frac{17}{5} . . . . . (i i i) 5 u + v = 2 . . . . . (i v)$
Multiplying (iv) by 3 and subtracting from(iii), we get
$2 u - 15 u = \frac{17}{5} - 6 \Rightarrow - 13 u = \frac{- 13}{5} \Rightarrow u = \frac{1}{5} \Rightarrow 3 x + 2 y = 5 (∵ \frac{1}{3 x + 2 y} = u) . . . . . (v)$
Now, substituting $u = \frac{1}{5}$ in (iv), we get
$1 + v = 2 \Rightarrow v = 1 \Rightarrow 3 x - 2 y = 1 (∵ \frac{1}{3 x - 2 y} = v) . . . . . (v i)$
Adding (v) and (vi), we get
$6 x = 6 \Rightarrow x = 1$
Substituting x = 1 in (v), we have
$3 + 2 y = 5 \Rightarrow y = 1$
Hence, x = 1 and y = 1.

Question 39:

The given equations are
$\frac{2}{3 x + 2 y} + \frac{3}{3 x - 2 y} = \frac{17}{5} . . . . . (i) \frac{5}{3 x + 2 y} + \frac{1}{3 x - 2 y} = 2 . . . . . (i i)$
Substituting $\frac{1}{3 x + 2 y} = u and \frac{1}{3 x - 2 y} = v$ in (i) and (ii), we get
$2 u + 3 v = \frac{17}{5} . . . . . (i i i) 5 u + v = 2 . . . . . (i v)$
Multiplying (iv) by 3 and subtracting from(iii), we get
$2 u - 15 u = \frac{17}{5} - 6 \Rightarrow - 13 u = \frac{- 13}{5} \Rightarrow u = \frac{1}{5} \Rightarrow 3 x + 2 y = 5 (∵ \frac{1}{3 x + 2 y} = u) . . . . . (v)$
Now, substituting $u = \frac{1}{5}$ in (iv), we get
$1 + v = 2 \Rightarrow v = 1 \Rightarrow 3 x - 2 y = 1 (∵ \frac{1}{3 x - 2 y} = v) . . . . . (v i)$
Adding (v) and (vi), we get
$6 x = 6 \Rightarrow x = 1$
Substituting x = 1 in (v), we have
$3 + 2 y = 5 \Rightarrow y = 1$
Hence, x = 1 and y = 1.

Answer:

The given equations can be written as
$\frac{3}{x} + \frac{6}{y} = 7 . . . . . (i) \frac{9}{x} + \frac{3}{y} = 11 . . . . . (i i)$
Multiplying (i) by 3 and subtracting (ii) from it, we get
$\frac{18}{y} - \frac{3}{y} = 21 - 11 \Rightarrow \frac{15}{y} = 10 \Rightarrow y = \frac{15}{10} = \frac{3}{2}$
Substituting $y = \frac{3}{2}$ in (i), we have
$\frac{3}{x} + \frac{6 \times 2}{3} = 7 \Rightarrow \frac{3}{x} = 7 - 4 = 3 \Rightarrow x = 1$
Hence, $x = 1 and y = \frac{3}{2}$ .

Question 40:

The given equations can be written as
$\frac{3}{x} + \frac{6}{y} = 7 . . . . . (i) \frac{9}{x} + \frac{3}{y} = 11 . . . . . (i i)$
Multiplying (i) by 3 and subtracting (ii) from it, we get
$\frac{18}{y} - \frac{3}{y} = 21 - 11 \Rightarrow \frac{15}{y} = 10 \Rightarrow y = \frac{15}{10} = \frac{3}{2}$
Substituting $y = \frac{3}{2}$ in (i), we have
$\frac{3}{x} + \frac{6 \times 2}{3} = 7 \Rightarrow \frac{3}{x} = 7 - 4 = 3 \Rightarrow x = 1$
Hence, $x = 1 and y = \frac{3}{2}$ .

Answer:

The given equations are
$x + y = a + b . . . . . (i) a x - b y = a^{2} - b^{2} . . . . . (i i)$
Multiplying (i) by b and adding it with (ii), we get
$b x + a x = a b + b^{2} + a^{2} - b^{2} \Rightarrow x = \frac{a b + a^{2}}{a + b} = a$
Substituting x = a in (i), we have
$a + y = a + b \Rightarrow y = b$
Hence, x = a and y = b.

Question 41:

The given equations are
$x + y = a + b . . . . . (i) a x - b y = a^{2} - b^{2} . . . . . (i i)$
Multiplying (i) by b and adding it with (ii), we get
$b x + a x = a b + b^{2} + a^{2} - b^{2} \Rightarrow x = \frac{a b + a^{2}}{a + b} = a$
Substituting x = a in (i), we have
$a + y = a + b \Rightarrow y = b$
Hence, x = a and y = b.

Answer:

The given equations are:
$\frac{x}{a} + \frac{y}{b} = 2$

⇒ $\frac{b x + a y}{a b} = 2$ [Taking LCM]
⇒bx + ay = 2ab .......(i)

Again, ax − by = (a² − b²) ........(ii)

On multiplying (i) by b and (ii) by a, we get:
b²x + bay = 2ab².........(iii)
a²x − bay = a(a² − b²) ........(iv)

On adding (iii) from (iv), we get:
(b² + a²)x = 2a²b + a(a² − b²)
⇒ (b² + a²)x = 2ab² + a³ − ab²
⇒ (b² + a²)x = ab²+ a³
⇒ (b² + a²)x = a(b² + a²)
⇒ $x = \frac{a (b^{2} + a^{2})}{(b^{2} + a^{2})} = a$
On substituting x = a in (i), we get:
ba + ay = 2ab
⇒ ay = ab
⇒ y = b

Hence, the solution is x = a and y = b.

Question 42:

The given equations are:
$\frac{x}{a} + \frac{y}{b} = 2$

⇒ $\frac{b x + a y}{a b} = 2$ [Taking LCM]
⇒bx + ay = 2ab .......(i)

Again, ax − by = (a² − b²) ........(ii)

On multiplying (i) by b and (ii) by a, we get:
b²x + bay = 2ab².........(iii)
a²x − bay = a(a² − b²) ........(iv)

On adding (iii) from (iv), we get:
(b² + a²)x = 2a²b + a(a² − b²)
⇒ (b² + a²)x = 2ab² + a³ − ab²
⇒ (b² + a²)x = ab²+ a³
⇒ (b² + a²)x = a(b² + a²)
⇒ $x = \frac{a (b^{2} + a^{2})}{(b^{2} + a^{2})} = a$
On substituting x = a in (i), we get:
ba + ay = 2ab
⇒ ay = ab
⇒ y = b

Hence, the solution is x = a and y = b.

Answer:

The given equations are
$p x + q y = p - q . . . . . (i) q x - p y = p + q . . . . . (i i)$
Multiplying (i) by p and (ii) by q and adding them, we get
$p^{2} x + q^{2} x = p^{2} - p q + p q + q^{2} \Rightarrow x = \frac{p^{2} + q^{2}}{p^{2} + q^{2}} = 1$
Substituting x = 1 in (i), we have
$p + q y = p - q \Rightarrow q y = - q \Rightarrow y = - 1$
Hence, x = 1 and $y = - 1$ .

Question 43:

The given equations are
$p x + q y = p - q . . . . . (i) q x - p y = p + q . . . . . (i i)$
Multiplying (i) by p and (ii) by q and adding them, we get
$p^{2} x + q^{2} x = p^{2} - p q + p q + q^{2} \Rightarrow x = \frac{p^{2} + q^{2}}{p^{2} + q^{2}} = 1$
Substituting x = 1 in (i), we have
$p + q y = p - q \Rightarrow q y = - q \Rightarrow y = - 1$
Hence, x = 1 and $y = - 1$ .

Answer:

The given equations are
$\frac{x}{a} - \frac{y}{b} = 0 . . . . . (i) a x + b y = a^{2} + b^{2} . . . . . (i i)$
From (i)
$y = \frac{b x}{a}$
Substituting $y = \frac{b x}{a}$ in (ii), we get
$a x + \frac{b \times b x}{a} = a^{2} + b^{2} \Rightarrow x = \frac{(a^{2} + b^{2}) \times a}{a^{2} + b^{2}} = a$
Now, substitute x = a in (ii) to get
$a^{2} + b y = a^{2} + b^{2} \Rightarrow b y = b^{2} \Rightarrow y = b$
Hence, x = a and y = b.

Question 44:

The given equations are
$\frac{x}{a} - \frac{y}{b} = 0 . . . . . (i) a x + b y = a^{2} + b^{2} . . . . . (i i)$
From (i)
$y = \frac{b x}{a}$
Substituting $y = \frac{b x}{a}$ in (ii), we get
$a x + \frac{b \times b x}{a} = a^{2} + b^{2} \Rightarrow x = \frac{(a^{2} + b^{2}) \times a}{a^{2} + b^{2}} = a$
Now, substitute x = a in (ii) to get
$a^{2} + b y = a^{2} + b^{2} \Rightarrow b y = b^{2} \Rightarrow y = b$
Hence, x = a and y = b.

Answer:

The given equations are:
6(ax + by) = 3a + 2b
⇒ 6ax + 6by = 3a + 2b ...............(i)

and 6(bx − ay) = 3b − 2a
⇒ 6bx − 6ay = 3b − 2a ...................(ii)

On multiplying (i) by a and (ii) by b, we get:
6a²x + 6aby = 3a² + 2ab ................(iii)
6b²x − 6aby = 3b² − 2ab ....................(iv)

On adding (iii) and (iv), we get:
6(a² + b²)x = 3(a² + b²)
$\Rightarrow x = \frac{3 (a^{2} + b^{2})}{6 (a^{2} + b^{2})} = \frac{1}{2}$

On substituting $x = \frac{1}{2}$ in (i), we get:
$6 a \times \frac{1}{2} + 6 b y = 3 a + 2 b$
⇒ 3a + 6by = 3a + 2b
⇒ 6by = 2b
⇒ y = $\frac{2 b}{6 b} = \frac{1}{3}$
Hence, the required solution is $x = \frac{1}{2}$ and $y = \frac{1}{3}$ .

Question 45:

The given equations are:
6(ax + by) = 3a + 2b
⇒ 6ax + 6by = 3a + 2b ...............(i)

and 6(bx − ay) = 3b − 2a
⇒ 6bx − 6ay = 3b − 2a ...................(ii)

On multiplying (i) by a and (ii) by b, we get:
6a²x + 6aby = 3a² + 2ab ................(iii)
6b²x − 6aby = 3b² − 2ab ....................(iv)

On adding (iii) and (iv), we get:
6(a² + b²)x = 3(a² + b²)
$\Rightarrow x = \frac{3 (a^{2} + b^{2})}{6 (a^{2} + b^{2})} = \frac{1}{2}$

On substituting $x = \frac{1}{2}$ in (i), we get:
$6 a \times \frac{1}{2} + 6 b y = 3 a + 2 b$
⇒ 3a + 6by = 3a + 2b
⇒ 6by = 2b
⇒ y = $\frac{2 b}{6 b} = \frac{1}{3}$
Hence, the required solution is $x = \frac{1}{2}$ and $y = \frac{1}{3}$ .

Answer:

The given equations are
$a x - b y = a^{2} + b^{2} . . . . . (i) x + y = 2 a . . . . . (i i)$
From (ii)
$y = 2 a - x$
Substituting $y = 2 a - x$ in (i), we get
$a x - b (2 a - x) = a^{2} + b^{2} \Rightarrow a x - 2 a b + b x = a^{2} + b^{2} \Rightarrow x = \frac{a^{2} + b^{2} + 2 a b}{a + b} = \frac{{(a + b)}^{2}}{a + b} = a + b$
Now, substitute x = a + b in (ii) to get
$a + b + y = 2 a \Rightarrow y = a - b$
Hence, $x = a + b and y = a - b$ .

Question 46:

The given equations are
$a x - b y = a^{2} + b^{2} . . . . . (i) x + y = 2 a . . . . . (i i)$
From (ii)
$y = 2 a - x$
Substituting $y = 2 a - x$ in (i), we get
$a x - b (2 a - x) = a^{2} + b^{2} \Rightarrow a x - 2 a b + b x = a^{2} + b^{2} \Rightarrow x = \frac{a^{2} + b^{2} + 2 a b}{a + b} = \frac{{(a + b)}^{2}}{a + b} = a + b$
Now, substitute x = a + b in (ii) to get
$a + b + y = 2 a \Rightarrow y = a - b$
Hence, $x = a + b and y = a - b$ .

Answer:

The given equations are:
$\frac{b x}{a} - \frac{a y}{b} + a + b = 0$
By taking LCM, we get:
b²x − a²y = −a²b − b²a .......(i)

and bx − ay + 2ab = 0
bx − ay = −2ab ........(ii)

On multiplying (ii) by a, we get:
abx − a²y = −2a²b .......(iii)

On subtracting (i) from (iii), we get:
abx − b²x = − 2a²b + a²b + b²a = −a²b + b²a
⇒ x(ab − b²) = −ab(a − b)
⇒ x(a − b)b = −ab(a − b)
∴ $x = \frac{- a b (a - b)}{(a - b) b} = - a$

On substituting x = −a in (i), we get:
b²(−a) − a²y = −a²b − b²a
⇒ −b²a − a²y = −a²b − b²a
⇒ −a²y = −a²b
⇒ y = b

Hence, the solution is x = −a and y = b.

Question 47:

The given equations are:
$\frac{b x}{a} - \frac{a y}{b} + a + b = 0$
By taking LCM, we get:
b²x − a²y = −a²b − b²a .......(i)

and bx − ay + 2ab = 0
bx − ay = −2ab ........(ii)

On multiplying (ii) by a, we get:
abx − a²y = −2a²b .......(iii)

On subtracting (i) from (iii), we get:
abx − b²x = − 2a²b + a²b + b²a = −a²b + b²a
⇒ x(ab − b²) = −ab(a − b)
⇒ x(a − b)b = −ab(a − b)
∴ $x = \frac{- a b (a - b)}{(a - b) b} = - a$

On substituting x = −a in (i), we get:
b²(−a) − a²y = −a²b − b²a
⇒ −b²a − a²y = −a²b − b²a
⇒ −a²y = −a²b
⇒ y = b

Hence, the solution is x = −a and y = b.

Answer:

The given equations are:

$\frac{b x}{a} + \frac{a y}{b} = a^{2} + b^{2}$
By taking LCM, we get:
$\frac{b^{2} x + a^{2} y}{a b} = a^{2} + b^{2}$
⇒ b²x + a²y = (ab)a² + b²
⇒ b²x + a²y = a³b + ab³ .......(i)

Also, x + y = 2ab........(ii)

On multiplying (ii) by a², we get:
a²x + a²y = 2a³b.........(iii)

On subtracting (iii) from (i), we get:
(b² − a²)x = a³b + ab³ − 2a³b
⇒ (b² − a²)x = −a³b + ab³
⇒ (b² − a²)x = ab(b² − a²)
⇒ (b² − a²)x = ab(b² − a²)
⇒ $x = \frac{a b (b^{2} - a^{2})}{(b^{2} - a^{2})} = a b$

On substituting x = ab in (i), we get:
b²(ab) + a²y = a³b + ab³
⇒ a²y = a³b
⇒ $\frac{a^{3} b}{a^{2}} = a b$
Hence, the solution is x = ab and y = ab.

Question 48:

The given equations are:

$\frac{b x}{a} + \frac{a y}{b} = a^{2} + b^{2}$
By taking LCM, we get:
$\frac{b^{2} x + a^{2} y}{a b} = a^{2} + b^{2}$
⇒ b²x + a²y = (ab)a² + b²
⇒ b²x + a²y = a³b + ab³ .......(i)

Also, x + y = 2ab........(ii)

On multiplying (ii) by a², we get:
a²x + a²y = 2a³b.........(iii)

On subtracting (iii) from (i), we get:
(b² − a²)x = a³b + ab³ − 2a³b
⇒ (b² − a²)x = −a³b + ab³
⇒ (b² − a²)x = ab(b² − a²)
⇒ (b² − a²)x = ab(b² − a²)
⇒ $x = \frac{a b (b^{2} - a^{2})}{(b^{2} - a^{2})} = a b$

On substituting x = ab in (i), we get:
b²(ab) + a²y = a³b + ab³
⇒ a²y = a³b
⇒ $\frac{a^{3} b}{a^{2}} = a b$
Hence, the solution is x = ab and y = ab.

Answer:

The given equations are
$x + y = a + b . . . . . (i) a x - b y = a^{2} - b^{2} . . . . . (i i)$
From (i)
$y = a + b - x$
Substituting $y = a + b - x$ in (ii), we get
$a x - b (a + b - x) = a^{2} - b^{2} \Rightarrow a x - a b - b^{2} + b x = a^{2} - b^{2} \Rightarrow x = \frac{a^{2} + a b}{a + b} = a$
Now, substitute x = a in (i) to get
$a + y = a + b \Rightarrow y = b$
Hence, x = a and y = b.

Question 49:

The given equations are
$x + y = a + b . . . . . (i) a x - b y = a^{2} - b^{2} . . . . . (i i)$
From (i)
$y = a + b - x$
Substituting $y = a + b - x$ in (ii), we get
$a x - b (a + b - x) = a^{2} - b^{2} \Rightarrow a x - a b - b^{2} + b x = a^{2} - b^{2} \Rightarrow x = \frac{a^{2} + a b}{a + b} = a$
Now, substitute x = a in (i) to get
$a + y = a + b \Rightarrow y = b$
Hence, x = a and y = b.

Answer:

The given equations are
$a^{2} x + b^{2} y = c^{2} . . . . . (i) b^{2} x + a^{2} y = d^{2} . . . . . (i i)$
Multiplying (i) by a²and (ii) by b² and subtracting, we get
$a^{4} x - b^{4} x = a^{2} c^{2} - b^{2} d^{2} \Rightarrow x = \frac{a^{2} c^{2} - b^{2} d^{2}}{a^{4} - b^{4}}$
Now, multiplying (i) by b²and (ii) by a² and subtracting, we get
$b^{4} y - a^{4} y = b^{2} c^{2} - a^{2} d^{2} \Rightarrow y = \frac{b^{2} c^{2} - a^{2} d^{2}}{b^{4} - a^{4}}$

Hence, $x = \frac{a^{2} c^{2} - b^{2} d^{2}}{a^{4} - b^{4}} and y = \frac{b^{2} c^{2} - a^{2} d^{2}}{b^{4} - a^{4}}$ .

Question 50:

The given equations are
$a^{2} x + b^{2} y = c^{2} . . . . . (i) b^{2} x + a^{2} y = d^{2} . . . . . (i i)$
Multiplying (i) by a²and (ii) by b² and subtracting, we get
$a^{4} x - b^{4} x = a^{2} c^{2} - b^{2} d^{2} \Rightarrow x = \frac{a^{2} c^{2} - b^{2} d^{2}}{a^{4} - b^{4}}$
Now, multiplying (i) by b²and (ii) by a² and subtracting, we get
$b^{4} y - a^{4} y = b^{2} c^{2} - a^{2} d^{2} \Rightarrow y = \frac{b^{2} c^{2} - a^{2} d^{2}}{b^{4} - a^{4}}$

Hence, $x = \frac{a^{2} c^{2} - b^{2} d^{2}}{a^{4} - b^{4}} and y = \frac{b^{2} c^{2} - a^{2} d^{2}}{b^{4} - a^{4}}$ .

Answer:

The given equations are
$\frac{x}{a} + \frac{y}{b} = a + b . . . . . (i) \frac{x}{a^{2}} + \frac{y}{b^{2}} = 2 . . . . . (i i)$
Multiplying (i) by band (ii) by b² and subtracting, we get
$\frac{b x}{a} - \frac{b^{2} x}{a^{2}} = a b + b^{2} - 2 b^{2} \Rightarrow \frac{a b - b^{2}}{a^{2}} x = a b - b^{2} \Rightarrow x = \frac{(a b - b^{2}) a^{2}}{a b - b^{2}} = a^{2}$
Now, substituting x = a²in (i), we get
$\frac{a^{2}}{a} + \frac{y}{b} = a + b \Rightarrow \frac{y}{b} = a + b - a = b \Rightarrow y = b^{2}$

Hence, $x = a^{2} and y = b^{2}$ .

Question 1:

The given equations are
$\frac{x}{a} + \frac{y}{b} = a + b . . . . . (i) \frac{x}{a^{2}} + \frac{y}{b^{2}} = 2 . . . . . (i i)$
Multiplying (i) by band (ii) by b² and subtracting, we get
$\frac{b x}{a} - \frac{b^{2} x}{a^{2}} = a b + b^{2} - 2 b^{2} \Rightarrow \frac{a b - b^{2}}{a^{2}} x = a b - b^{2} \Rightarrow x = \frac{(a b - b^{2}) a^{2}}{a b - b^{2}} = a^{2}$
Now, substituting x = a²in (i), we get
$\frac{a^{2}}{a} + \frac{y}{b} = a + b \Rightarrow \frac{y}{b} = a + b - a = b \Rightarrow y = b^{2}$

Hence, $x = a^{2} and y = b^{2}$ .

Answer:

The given equations are:
x + 2y + 1 = 0 ...(i)
2x − 3y − 12 = 0 ...(ii)
Here, a₁ = 1, b₁= 2, c₁ = 1, a₂ = 2, b₂ = −3 and c₂ = −12
By cross multiplication, we have:

∴ $\frac{x}{[2 \times (- 12) - 1 \times (- 3)]} = \frac{y}{[1 \times 2 - 1 \times (- 12)]} = \frac{1}{[1 \times (- 3) - 2 \times 2]}$
⇒ $\frac{x}{(- 24 + 3)} = \frac{y}{(2 + 12)} = \frac{1}{(- 3 - 4)}$
⇒ $\frac{x}{(- 21)} = \frac{y}{(14)} = \frac{1}{(- 7)}$
⇒ $x = \frac{- 21}{- 7} = 3, y = \frac{14}{- 7} = - 2$
Hence, x = 3 and y = −2 is the required solution.

Question 2:

The given equations are:
x + 2y + 1 = 0 ...(i)
2x − 3y − 12 = 0 ...(ii)
Here, a₁ = 1, b₁= 2, c₁ = 1, a₂ = 2, b₂ = −3 and c₂ = −12
By cross multiplication, we have:

∴ $\frac{x}{[2 \times (- 12) - 1 \times (- 3)]} = \frac{y}{[1 \times 2 - 1 \times (- 12)]} = \frac{1}{[1 \times (- 3) - 2 \times 2]}$
⇒ $\frac{x}{(- 24 + 3)} = \frac{y}{(2 + 12)} = \frac{1}{(- 3 - 4)}$
⇒ $\frac{x}{(- 21)} = \frac{y}{(14)} = \frac{1}{(- 7)}$
⇒ $x = \frac{- 21}{- 7} = 3, y = \frac{14}{- 7} = - 2$
Hence, x = 3 and y = −2 is the required solution.

Answer:

The given equations are:
3x − 2y + 3 = 0 ...(i)
4x + 3y − 47 = 0 ...(ii)
Here, a₁ = 3, b₁= −2 , c₁ = 3, a₂ = 4, b₂ = 3 and c₂ = −47
By cross multiplication, we have:

∴ $\frac{x}{[(- 2) \times (- 47) - 3 \times 3]} = \frac{y}{[3 \times 4 - (- 47) \times 3]} = \frac{1}{[3 \times 3 - (- 2) \times 4]}$
⇒ $\frac{x}{(94 - 9)} = \frac{y}{(12 + 141)} = \frac{z}{(9 + 8)}$
⇒ $\frac{x}{85} = \frac{y}{153} = \frac{1}{17}$
⇒ $x = \frac{85}{17} = 5, y = \frac{153}{17} = 9$
Hence, x = 5 and y = 9 is the required solution.

Question 3:

The given equations are:
3x − 2y + 3 = 0 ...(i)
4x + 3y − 47 = 0 ...(ii)
Here, a₁ = 3, b₁= −2 , c₁ = 3, a₂ = 4, b₂ = 3 and c₂ = −47
By cross multiplication, we have:

∴ $\frac{x}{[(- 2) \times (- 47) - 3 \times 3]} = \frac{y}{[3 \times 4 - (- 47) \times 3]} = \frac{1}{[3 \times 3 - (- 2) \times 4]}$
⇒ $\frac{x}{(94 - 9)} = \frac{y}{(12 + 141)} = \frac{z}{(9 + 8)}$
⇒ $\frac{x}{85} = \frac{y}{153} = \frac{1}{17}$
⇒ $x = \frac{85}{17} = 5, y = \frac{153}{17} = 9$
Hence, x = 5 and y = 9 is the required solution.

Answer:

The given equations are:
6x − 5y − 16 = 0 ...(i)
7x − 13y + 10 = 0 ...(ii)
Here, a₁ = 6, b₁= −5 , c₁ = −16, a₂ = 7, b₂ = −13 and c₂ = 10
By cross multiplication, we have:

∴ $\frac{x}{[(- 5) \times 10 - (- 16) \times (- 13)]} = \frac{y}{[(- 16) \times 7 - 10 \times 6]} = \frac{1}{[6 \times (- 13) - (- 5) \times 7]}$
⇒ $\frac{x}{(- 50 - 208)} = \frac{y}{(- 112 - 60)} = \frac{z}{(- 78 + 35)}$
⇒ $\frac{x}{(- 258)} = \frac{y}{(- 172)} = \frac{1}{(- 43)}$
⇒ $x = \frac{- 258}{- 43} = 6, y = \frac{- 172}{- 43} = 4$
Hence, x = 6 and y = 4 is the required solution.

Question 4:

The given equations are:
6x − 5y − 16 = 0 ...(i)
7x − 13y + 10 = 0 ...(ii)
Here, a₁ = 6, b₁= −5 , c₁ = −16, a₂ = 7, b₂ = −13 and c₂ = 10
By cross multiplication, we have:

∴ $\frac{x}{[(- 5) \times 10 - (- 16) \times (- 13)]} = \frac{y}{[(- 16) \times 7 - 10 \times 6]} = \frac{1}{[6 \times (- 13) - (- 5) \times 7]}$
⇒ $\frac{x}{(- 50 - 208)} = \frac{y}{(- 112 - 60)} = \frac{z}{(- 78 + 35)}$
⇒ $\frac{x}{(- 258)} = \frac{y}{(- 172)} = \frac{1}{(- 43)}$
⇒ $x = \frac{- 258}{- 43} = 6, y = \frac{- 172}{- 43} = 4$
Hence, x = 6 and y = 4 is the required solution.

Answer:

The given equations are:
3x + 2y + 25 = 0 ...(i)
2x + y + 10 = 0 ...(ii)
Here, a₁ = 3, b₁= 2 , c₁ = 25, a₂ = 2, b₂ = 1 and c₂ = 10
By cross multiplication, we have:

∴ $\frac{x}{[2 \times 10 - 25 \times 1]} = \frac{y}{[25 \times 2 - 10 \times 3]} = \frac{1}{[3 \times 1 - 2 \times 2]}$
⇒ $\frac{x}{(20 - 25)} = \frac{y}{(50 - 30)} = \frac{1}{(3 - 4)}$
⇒ $\frac{x}{(- 5)} = \frac{y}{20} = \frac{1}{(- 1)}$
⇒ $x = \frac{- 5}{- 1} = 5, y = \frac{20}{(- 1)} = - 20$
Hence, x = 5 and y = −20 is the required solution.

Question 5:

The given equations are:
3x + 2y + 25 = 0 ...(i)
2x + y + 10 = 0 ...(ii)
Here, a₁ = 3, b₁= 2 , c₁ = 25, a₂ = 2, b₂ = 1 and c₂ = 10
By cross multiplication, we have:

∴ $\frac{x}{[2 \times 10 - 25 \times 1]} = \frac{y}{[25 \times 2 - 10 \times 3]} = \frac{1}{[3 \times 1 - 2 \times 2]}$
⇒ $\frac{x}{(20 - 25)} = \frac{y}{(50 - 30)} = \frac{1}{(3 - 4)}$
⇒ $\frac{x}{(- 5)} = \frac{y}{20} = \frac{1}{(- 1)}$
⇒ $x = \frac{- 5}{- 1} = 5, y = \frac{20}{(- 1)} = - 20$
Hence, x = 5 and y = −20 is the required solution.

Answer:

The given equations may be written as:
2x + 5y − 1 = 0 ...(i)
2x + 3y − 3 = 0 ...(ii)
Here, a₁ = 2, b₁= 5, c₁ = −1, a₂ = 2, b₂ = 3 and c₂ = −3
By cross multiplication, we have:

∴ $\frac{x}{[5 \times (- 3) - 3 \times (- 1)]} = \frac{y}{[(- 1) \times 2 - (- 3) \times 2]} = \frac{1}{[2 \times 3 - 2 \times 5]}$
⇒ $\frac{x}{(- 15 + 3)} = \frac{y}{(- 2 + 6)} = \frac{z}{(6 - 10)}$
⇒ $\frac{x}{- 12} = \frac{y}{4} = \frac{1}{- 4}$
⇒ $x = \frac{- 12}{- 4} = 3, y = \frac{4}{- 4} = - 1$
Hence, x = 3 and y = −1 is the required solution.

Question 6:

The given equations may be written as:
2x + 5y − 1 = 0 ...(i)
2x + 3y − 3 = 0 ...(ii)
Here, a₁ = 2, b₁= 5, c₁ = −1, a₂ = 2, b₂ = 3 and c₂ = −3
By cross multiplication, we have:

∴ $\frac{x}{[5 \times (- 3) - 3 \times (- 1)]} = \frac{y}{[(- 1) \times 2 - (- 3) \times 2]} = \frac{1}{[2 \times 3 - 2 \times 5]}$
⇒ $\frac{x}{(- 15 + 3)} = \frac{y}{(- 2 + 6)} = \frac{z}{(6 - 10)}$
⇒ $\frac{x}{- 12} = \frac{y}{4} = \frac{1}{- 4}$
⇒ $x = \frac{- 12}{- 4} = 3, y = \frac{4}{- 4} = - 1$
Hence, x = 3 and y = −1 is the required solution.

Answer:

The given equations may be written as:
2x + y − 35 = 0 ...(i)
3x + 4y − 65 = 0 ...(ii)
Here, a₁ = 2, b₁= 1, c₁ = −35, a₂ = 3, b₂ = 4 and c₂ = −65
By cross multiplication, we have:

∴ $\frac{x}{[1 \times (- 65) - 4 \times (- 35)]} = \frac{y}{[(- 35) \times 3 - (- 65) \times 2]} = \frac{1}{[2 \times 4 - 3 \times 1]}$
⇒ $\frac{x}{(- 65 + 140)} = \frac{y}{(- 105 + 130)} = \frac{1}{(8 - 3)}$
⇒ $\frac{x}{75} = \frac{y}{25} = \frac{1}{5}$
⇒ $x = \frac{75}{5} = 15, y = \frac{25}{5} = 5$
Hence, x = 15 and y = 5 is the required solution.

Question 7:

The given equations may be written as:
2x + y − 35 = 0 ...(i)
3x + 4y − 65 = 0 ...(ii)
Here, a₁ = 2, b₁= 1, c₁ = −35, a₂ = 3, b₂ = 4 and c₂ = −65
By cross multiplication, we have:

∴ $\frac{x}{[1 \times (- 65) - 4 \times (- 35)]} = \frac{y}{[(- 35) \times 3 - (- 65) \times 2]} = \frac{1}{[2 \times 4 - 3 \times 1]}$
⇒ $\frac{x}{(- 65 + 140)} = \frac{y}{(- 105 + 130)} = \frac{1}{(8 - 3)}$
⇒ $\frac{x}{75} = \frac{y}{25} = \frac{1}{5}$
⇒ $x = \frac{75}{5} = 15, y = \frac{25}{5} = 5$
Hence, x = 15 and y = 5 is the required solution.

Answer:

The given equations may be written as:
7x − 2y − 3 = 0 ...(i)
$22 x - 3 y - 16 = 0$ ...(ii)
Here, a₁ = 7, b₁= −2 , c₁ = −3, a₂ = 22, b₂ = $- 3$ and c₂ = −16
By cross multiplication, we have:

∴ $\frac{x}{[(- 2) \times (- 16) - (- 3) \times (- 3)]} = \frac{y}{[(- 3) \times 22 - (- 16) \times 7]} = \frac{1}{[7 \times (- 3) - 22 \times (- 2)]}$
⇒ $\frac{x}{(32 - 9)} = \frac{y}{(- 66 + 112)} = \frac{1}{(- 21 + 44)}$
⇒ $\frac{x}{(23)} = \frac{y}{46} = \frac{1}{(23)}$
⇒ $x = \frac{23}{23} = 1, y = \frac{46}{23} = 2$
Hence, x = 1 and y = 2 is the required solution.

Question 8:

The given equations may be written as:
7x − 2y − 3 = 0 ...(i)
$22 x - 3 y - 16 = 0$ ...(ii)
Here, a₁ = 7, b₁= −2 , c₁ = −3, a₂ = 22, b₂ = $- 3$ and c₂ = −16
By cross multiplication, we have:

∴ $\frac{x}{[(- 2) \times (- 16) - (- 3) \times (- 3)]} = \frac{y}{[(- 3) \times 22 - (- 16) \times 7]} = \frac{1}{[7 \times (- 3) - 22 \times (- 2)]}$
⇒ $\frac{x}{(32 - 9)} = \frac{y}{(- 66 + 112)} = \frac{1}{(- 21 + 44)}$
⇒ $\frac{x}{(23)} = \frac{y}{46} = \frac{1}{(23)}$
⇒ $x = \frac{23}{23} = 1, y = \frac{46}{23} = 2$
Hence, x = 1 and y = 2 is the required solution.

Answer:

The given equations may be written as:
$\frac{x}{6} + \frac{y}{15} - 4 = 0$ ...(i)
$\frac{x}{3} - \frac{y}{12} - \frac{19}{4} = 0$ ...(ii)
Here, $a_{1} = \frac{1}{6}, b_{1} = \frac{1}{15}, c_{1} = - 4, a_{2} = \frac{1}{3}, b_{2} = - \frac{1}{12} and c_{2} = - \frac{19}{4}$
By cross multiplication, we have:

∴ $\frac{x}{[\frac{1}{15} \times (- \frac{19}{4}) - (- \frac{1}{12}) \times (- 4)]} = \frac{y}{[(- 4) \times \frac{1}{3} - (\frac{1}{6}) \times (- \frac{19}{4})]} = \frac{1}{[\frac{1}{6} \times (- \frac{1}{12}) - \frac{1}{3} \times \frac{1}{15}]}$
⇒ $\frac{x}{(- \frac{19}{60} - \frac{1}{3})} = \frac{y}{(- \frac{4}{3} + \frac{19}{24})} = \frac{1}{(- \frac{1}{72} - \frac{1}{45})}$
⇒ $\frac{x}{(- \frac{39}{60})} = \frac{y}{(- \frac{13}{24})} = \frac{1}{(- \frac{13}{360})}$
⇒ $x = [(- \frac{39}{60}) \times (- \frac{360}{13})] = 18, y = [(- \frac{13}{24}) \times (- \frac{360}{13})] = 15$
Hence, x = 18 and y = 15 is the required solution.

Question 9:

The given equations may be written as:
$\frac{x}{6} + \frac{y}{15} - 4 = 0$ ...(i)
$\frac{x}{3} - \frac{y}{12} - \frac{19}{4} = 0$ ...(ii)
Here, $a_{1} = \frac{1}{6}, b_{1} = \frac{1}{15}, c_{1} = - 4, a_{2} = \frac{1}{3}, b_{2} = - \frac{1}{12} and c_{2} = - \frac{19}{4}$
By cross multiplication, we have:

∴ $\frac{x}{[\frac{1}{15} \times (- \frac{19}{4}) - (- \frac{1}{12}) \times (- 4)]} = \frac{y}{[(- 4) \times \frac{1}{3} - (\frac{1}{6}) \times (- \frac{19}{4})]} = \frac{1}{[\frac{1}{6} \times (- \frac{1}{12}) - \frac{1}{3} \times \frac{1}{15}]}$
⇒ $\frac{x}{(- \frac{19}{60} - \frac{1}{3})} = \frac{y}{(- \frac{4}{3} + \frac{19}{24})} = \frac{1}{(- \frac{1}{72} - \frac{1}{45})}$
⇒ $\frac{x}{(- \frac{39}{60})} = \frac{y}{(- \frac{13}{24})} = \frac{1}{(- \frac{13}{360})}$
⇒ $x = [(- \frac{39}{60}) \times (- \frac{360}{13})] = 18, y = [(- \frac{13}{24}) \times (- \frac{360}{13})] = 15$
Hence, x = 18 and y = 15 is the required solution.

Answer:

Taking $\frac{1}{x} = u$ and $\frac{1}{y} = v$ , the given equations become:
u + v = 7
2u + 3v = 17

The given equations may be written as:
u + v − 7 = 0            ...(i)
2u + 3v − 17 = 0      ...(ii)

Here, a₁= 1, b₁ = 1, c₁ = −7, a₂ = 2, b₂ = 3 and c₂ = −17
By cross multiplication, we have:

∴ $\frac{u}{[1 \times (- 17) - 3 \times (- 7)]} = \frac{v}{[(- 7) \times 2 - 1 \times (- 17)]} = \frac{1}{[3 - 2]}$
⇒ $\frac{u}{- 17 + 21} = \frac{v}{- 14 + 17} = \frac{1}{1}$
⇒ $\frac{u}{4} = \frac{v}{3} = \frac{1}{1}$
⇒ $u = \frac{4}{1} = 4, v = \frac{3}{1} = 3$
⇒ $\frac{1}{x} = 4, \frac{1}{y} = 3$
⇒ $x = \frac{1}{4}, y = \frac{1}{3}$
Hence, $x = \frac{1}{4}$ and $y = \frac{1}{3}$ is the required solution.

Question 10:

Taking $\frac{1}{x} = u$ and $\frac{1}{y} = v$ , the given equations become:
u + v = 7
2u + 3v = 17

The given equations may be written as:
u + v − 7 = 0            ...(i)
2u + 3v − 17 = 0      ...(ii)

Here, a₁= 1, b₁ = 1, c₁ = −7, a₂ = 2, b₂ = 3 and c₂ = −17
By cross multiplication, we have:

∴ $\frac{u}{[1 \times (- 17) - 3 \times (- 7)]} = \frac{v}{[(- 7) \times 2 - 1 \times (- 17)]} = \frac{1}{[3 - 2]}$
⇒ $\frac{u}{- 17 + 21} = \frac{v}{- 14 + 17} = \frac{1}{1}$
⇒ $\frac{u}{4} = \frac{v}{3} = \frac{1}{1}$
⇒ $u = \frac{4}{1} = 4, v = \frac{3}{1} = 3$
⇒ $\frac{1}{x} = 4, \frac{1}{y} = 3$
⇒ $x = \frac{1}{4}, y = \frac{1}{3}$
Hence, $x = \frac{1}{4}$ and $y = \frac{1}{3}$ is the required solution.

Answer:

Taking $\frac{1}{x + y} = u$ and $\frac{1}{x - y} = v$ , the given equations become:
5u − 2v + 1 = 0      ...(i)
15u + 7v − 10 = 0    ...(ii)
Here, a₁= 5, b₁ = −2, c₁ = 1, a₂ = 15, b₂ = −7 and c₂ = −10
By cross multiplication, we have:

∴ $\frac{u}{[- 2 \times (- 10) - 1 \times 7]} = \frac{v}{[1 \times 15 - (- 10) \times 5]} = \frac{1}{[35 + 30]}$
⇒ $\frac{u}{20 - 7} = \frac{v}{15 + 50} = \frac{1}{65}$
⇒ $\frac{u}{13} = \frac{v}{65} = \frac{1}{65}$
⇒ $u = \frac{13}{65} = \frac{1}{5}, v = \frac{65}{65} = 1$
⇒ $\frac{1}{x + y} = \frac{1}{5}, \frac{1}{x - y} = 1$
So, (x + y) = 5            ...(iii)
and (x − y) = 1           ...(iv)

Again, the above equations (iii) and (iv) may be written as:
x + y − 5 = 0           ...(v)
x − y − 1 = 0          ...(vi)
Here, a₁= 1, b₁ = 1, c₁ = −5, a₂ = 1, b₂ = −1 and c₂ = −1
By cross multiplication, we have:

∴ $\frac{x}{[1 \times (- 1) - (- 5) \times (- 1)]} = \frac{y}{[(- 5) \times 1 - (- 1) \times 1]} = \frac{1}{[1 \times (- 1) - 1 \times 1]}$

⇒ $\frac{x}{(- 1 - 5)} = \frac{y}{(- 5 + 1)} = \frac{1}{(- 1 - 1)}$
⇒ $\frac{x}{- 6} = \frac{y}{- 4} = \frac{1}{- 2}$
⇒ $x = \frac{- 6}{- 2} = 3, y = \frac{- 4}{- 2} = 2$
Hence, x = 3 and y = 2 is the required solution.

Question 11:

Taking $\frac{1}{x + y} = u$ and $\frac{1}{x - y} = v$ , the given equations become:
5u − 2v + 1 = 0      ...(i)
15u + 7v − 10 = 0    ...(ii)
Here, a₁= 5, b₁ = −2, c₁ = 1, a₂ = 15, b₂ = −7 and c₂ = −10
By cross multiplication, we have:

∴ $\frac{u}{[- 2 \times (- 10) - 1 \times 7]} = \frac{v}{[1 \times 15 - (- 10) \times 5]} = \frac{1}{[35 + 30]}$
⇒ $\frac{u}{20 - 7} = \frac{v}{15 + 50} = \frac{1}{65}$
⇒ $\frac{u}{13} = \frac{v}{65} = \frac{1}{65}$
⇒ $u = \frac{13}{65} = \frac{1}{5}, v = \frac{65}{65} = 1$
⇒ $\frac{1}{x + y} = \frac{1}{5}, \frac{1}{x - y} = 1$
So, (x + y) = 5            ...(iii)
and (x − y) = 1           ...(iv)

Again, the above equations (iii) and (iv) may be written as:
x + y − 5 = 0           ...(v)
x − y − 1 = 0          ...(vi)
Here, a₁= 1, b₁ = 1, c₁ = −5, a₂ = 1, b₂ = −1 and c₂ = −1
By cross multiplication, we have:

∴ $\frac{x}{[1 \times (- 1) - (- 5) \times (- 1)]} = \frac{y}{[(- 5) \times 1 - (- 1) \times 1]} = \frac{1}{[1 \times (- 1) - 1 \times 1]}$

⇒ $\frac{x}{(- 1 - 5)} = \frac{y}{(- 5 + 1)} = \frac{1}{(- 1 - 1)}$
⇒ $\frac{x}{- 6} = \frac{y}{- 4} = \frac{1}{- 2}$
⇒ $x = \frac{- 6}{- 2} = 3, y = \frac{- 4}{- 2} = 2$
Hence, x = 3 and y = 2 is the required solution.

Answer:

The given equations may be written as:
$\frac{a x}{b} - \frac{b y}{a} - (a + b) = 0$ ...(i)
$a x - b y - 2 a b = 0$ ...(ii)
Here, a₁= $\frac{a}{b}$ , b₁ = $\frac{- b}{a}$ , c₁ = −(a + b), a₂ = a, b₂ = −b and c₂ = −2ab
By cross multiplication, we have:

∴ $\frac{x}{(- \frac{b}{a}) \times (- 2 a b) - (- b) \times (- (a + b))} = \frac{y}{- (a + b) \times a - (- 2 a b) \times \frac{a}{b}} = \frac{1}{\frac{a}{b} \times (- b) - a \times (- \frac{b}{a})}$
⇒ $\frac{x}{2 b^{2} - b (a + b)} = \frac{y}{- a (a + b) + 2 a^{2}} = \frac{1}{- a + b}$
⇒ $\frac{x}{2 b^{2} - a b - b^{2}} = \frac{y}{- a^{2} - a b + 2 a^{2}} = \frac{1}{- a + b}$
⇒ $\frac{x}{b^{2} - a b} = \frac{y}{a^{2} - a b} = \frac{1}{- (a - b)}$
⇒ $\frac{x}{- b (a - b)} = \frac{y}{a (a - b)} = \frac{1}{- (a - b)}$
⇒ $x = \frac{- b (a - b)}{- (a - b)} = b, y = \frac{a (a - b)}{- (a - b)} = - a$
Hence, x = b and y = −a is the required solution.

Question 12:

The given equations may be written as:
$\frac{a x}{b} - \frac{b y}{a} - (a + b) = 0$ ...(i)
$a x - b y - 2 a b = 0$ ...(ii)
Here, a₁= $\frac{a}{b}$ , b₁ = $\frac{- b}{a}$ , c₁ = −(a + b), a₂ = a, b₂ = −b and c₂ = −2ab
By cross multiplication, we have:

∴ $\frac{x}{(- \frac{b}{a}) \times (- 2 a b) - (- b) \times (- (a + b))} = \frac{y}{- (a + b) \times a - (- 2 a b) \times \frac{a}{b}} = \frac{1}{\frac{a}{b} \times (- b) - a \times (- \frac{b}{a})}$
⇒ $\frac{x}{2 b^{2} - b (a + b)} = \frac{y}{- a (a + b) + 2 a^{2}} = \frac{1}{- a + b}$
⇒ $\frac{x}{2 b^{2} - a b - b^{2}} = \frac{y}{- a^{2} - a b + 2 a^{2}} = \frac{1}{- a + b}$
⇒ $\frac{x}{b^{2} - a b} = \frac{y}{a^{2} - a b} = \frac{1}{- (a - b)}$
⇒ $\frac{x}{- b (a - b)} = \frac{y}{a (a - b)} = \frac{1}{- (a - b)}$
⇒ $x = \frac{- b (a - b)}{- (a - b)} = b, y = \frac{a (a - b)}{- (a - b)} = - a$
Hence, x = b and y = −a is the required solution.

Answer:

The given equations may be written as:
2ax + 3by − (a + 2b) = 0 ...(i)
3ax + 2by − (2a + b) = 0 ...(ii)
Here, a₁= 2a, b₁ = 3b, c₁ = −(a + 2b), a₂ = 3a, b₂ = 2b and c₂ = −(2a + b)
By cross multiplication, we have:

∴ $\frac{x}{[3 b \times (- (2 a + b)) - 2 b \times (- (a + 2 b))]} = \frac{y}{[- (a + 2 b) \times 3 a - 2 a \times (- (2 a + b))]} = \frac{1}{[2 a \times 2 b - 3 a \times 3 b]}$
⇒ $\frac{x}{(- 6 a b - 3 b^{2} + 2 a b + 4 b^{2})} = \frac{y}{(- 3 a^{2} - 6 a b + 4 a^{2} + 2 a b)} = \frac{1}{4 a b - 9 a b}$
⇒ $\frac{x}{b^{2} - 4 a b} = \frac{y}{a^{2} - 4 a b} = \frac{1}{- 5 a b}$
⇒ $\frac{x}{- b (4 a - b)} = \frac{y}{- a (4 b - a)} = \frac{1}{- 5 a b}$
⇒ $x = \frac{- b (4 a - b)}{- 5 a b} = \frac{(4 a - b)}{5 a}, y = \frac{- a (4 b - a)}{- 5 a b} = \frac{(4 b - a)}{5 b}$
Hence, $x = \frac{(4 a - b)}{5 a}$ and $y = \frac{(4 b - a)}{5 b}$ is the required solution.

Question 13:

The given equations may be written as:
2ax + 3by − (a + 2b) = 0 ...(i)
3ax + 2by − (2a + b) = 0 ...(ii)
Here, a₁= 2a, b₁ = 3b, c₁ = −(a + 2b), a₂ = 3a, b₂ = 2b and c₂ = −(2a + b)
By cross multiplication, we have:

∴ $\frac{x}{[3 b \times (- (2 a + b)) - 2 b \times (- (a + 2 b))]} = \frac{y}{[- (a + 2 b) \times 3 a - 2 a \times (- (2 a + b))]} = \frac{1}{[2 a \times 2 b - 3 a \times 3 b]}$
⇒ $\frac{x}{(- 6 a b - 3 b^{2} + 2 a b + 4 b^{2})} = \frac{y}{(- 3 a^{2} - 6 a b + 4 a^{2} + 2 a b)} = \frac{1}{4 a b - 9 a b}$
⇒ $\frac{x}{b^{2} - 4 a b} = \frac{y}{a^{2} - 4 a b} = \frac{1}{- 5 a b}$
⇒ $\frac{x}{- b (4 a - b)} = \frac{y}{- a (4 b - a)} = \frac{1}{- 5 a b}$
⇒ $x = \frac{- b (4 a - b)}{- 5 a b} = \frac{(4 a - b)}{5 a}, y = \frac{- a (4 b - a)}{- 5 a b} = \frac{(4 b - a)}{5 b}$
Hence, $x = \frac{(4 a - b)}{5 a}$ and $y = \frac{(4 b - a)}{5 b}$ is the required solution.

Answer:

Substituting $\frac{1}{x} = u and \frac{1}{y} = v$ in the given equations, we get
$a u - b v + 0 = 0 . . . . . (i) a b^{2} u + a^{2} b v - (a^{2} + b^{2}) = 0 . . . . . (i i)$
Here, $a_{1} = a, b_{1} = - b, c_{1} = 0 and a_{2} = a b^{2}, b_{2} = a^{2} b, c_{2} = - (a^{2} + b^{2})$ .
So, by cross-multiplication, we have
$\frac{u}{b_{1} c_{2} - b_{2} c_{1}} = \frac{v}{c_{1} a_{2} - c_{2} a_{1}} = \frac{1}{a_{1} b_{2} - a_{2} b_{1}} \Rightarrow \frac{u}{(- b) [- (a^{2} + b^{2})] - (a^{2} b) (0)} = \frac{v}{(0) (a b^{2}) - (- a^{2} - b^{2}) (a)} = \frac{1}{(a) (a^{2} b) - (a b^{2}) (- b)} \Rightarrow \frac{u}{b (a^{2} + b^{2})} = \frac{v}{a (a^{2} + b^{2})} = \frac{1}{a b (a^{2} + b^{2})}$
$\Rightarrow u = \frac{b (a^{2} + b^{2})}{a b (a^{2} + b^{2})}, v = \frac{a (a^{2} + b^{2})}{a b (a^{2} + b^{2})} \Rightarrow u = \frac{1}{a}, v = \frac{1}{b} \Rightarrow \frac{1}{x} = \frac{1}{a}, \frac{1}{y} = \frac{1}{b} \Rightarrow x = a, y = b$
Hence, x = a and y = b.

Question 1:

Substituting $\frac{1}{x} = u and \frac{1}{y} = v$ in the given equations, we get
$a u - b v + 0 = 0 . . . . . (i) a b^{2} u + a^{2} b v - (a^{2} + b^{2}) = 0 . . . . . (i i)$
Here, $a_{1} = a, b_{1} = - b, c_{1} = 0 and a_{2} = a b^{2}, b_{2} = a^{2} b, c_{2} = - (a^{2} + b^{2})$ .
So, by cross-multiplication, we have
$\frac{u}{b_{1} c_{2} - b_{2} c_{1}} = \frac{v}{c_{1} a_{2} - c_{2} a_{1}} = \frac{1}{a_{1} b_{2} - a_{2} b_{1}} \Rightarrow \frac{u}{(- b) [- (a^{2} + b^{2})] - (a^{2} b) (0)} = \frac{v}{(0) (a b^{2}) - (- a^{2} - b^{2}) (a)} = \frac{1}{(a) (a^{2} b) - (a b^{2}) (- b)} \Rightarrow \frac{u}{b (a^{2} + b^{2})} = \frac{v}{a (a^{2} + b^{2})} = \frac{1}{a b (a^{2} + b^{2})}$
$\Rightarrow u = \frac{b (a^{2} + b^{2})}{a b (a^{2} + b^{2})}, v = \frac{a (a^{2} + b^{2})}{a b (a^{2} + b^{2})} \Rightarrow u = \frac{1}{a}, v = \frac{1}{b} \Rightarrow \frac{1}{x} = \frac{1}{a}, \frac{1}{y} = \frac{1}{b} \Rightarrow x = a, y = b$
Hence, x = a and y = b.

Answer:

The given system of equations is:
3x + 5y = 12
5x + 3y = 4
These equations are of the forms:
a₁x + b₁y + c₁= 0 and a₂x + b₂y + c₂ = 0
where, a₁ = 3, b₁= 5, c₁ = −12 and a₂ = 5, b₂= 3, c₂ = −4
For a unique solution, we must have:
$\frac{a_{1}}{a_{2}} \neq \frac{b_{1}}{b_{2}}$ , i.e., $\frac{3}{5} \neq \frac{5}{3}$
Hence, the given system of equations has a unique solution.

Again, the given equations are:
3x + 5y = 12            ...(i)
5x + 3y = 4              ...(ii)
On multiplying (i) by 3 and (ii) by 5, we get:
9x + 15y = 36          ...(iii)
25x + 15y = 20        ...(iv)
On subtracting (iii) from (iv), we get:
16x = −16
⇒ x = −1
On substituting x = −1 in (i), we get:
3(−1) + 5y = 12
⇒ 5y = (12 + 3) = 15
⇒ y = 3
Hence, x = −1 and y = 3 is the required solution.

Question 2:

The given system of equations is:
3x + 5y = 12
5x + 3y = 4
These equations are of the forms:
a₁x + b₁y + c₁= 0 and a₂x + b₂y + c₂ = 0
where, a₁ = 3, b₁= 5, c₁ = −12 and a₂ = 5, b₂= 3, c₂ = −4
For a unique solution, we must have:
$\frac{a_{1}}{a_{2}} \neq \frac{b_{1}}{b_{2}}$ , i.e., $\frac{3}{5} \neq \frac{5}{3}$
Hence, the given system of equations has a unique solution.

Again, the given equations are:
3x + 5y = 12            ...(i)
5x + 3y = 4              ...(ii)
On multiplying (i) by 3 and (ii) by 5, we get:
9x + 15y = 36          ...(iii)
25x + 15y = 20        ...(iv)
On subtracting (iii) from (iv), we get:
16x = −16
⇒ x = −1
On substituting x = −1 in (i), we get:
3(−1) + 5y = 12
⇒ 5y = (12 + 3) = 15
⇒ y = 3
Hence, x = −1 and y = 3 is the required solution.

Answer:

The system of equations can be written as
$2 x - 3 y - 17 = 0 . . . . . (i) 4 x + y - 13 = 0 . . . . . (i i)$
The given equations are of the form
$a_{1} x + b_{1} y + c_{1} = 0 and a_{2} x + b_{2} y + c_{2} = 0$
where $a_{1} = 2, b_{1} = - 3, c_{1} = - 17 and a_{2} = 4, b_{2} = 1, c_{2} = - 13$
Now,
$\frac{a_{1}}{a_{2}} = \frac{2}{4} = \frac{1}{2} and \frac{b_{1}}{b_{2}} = \frac{- 3}{1} = - 3$
Since, $\frac{a_{1}}{a_{2}} \neq \frac{b_{1}}{b_{2}}$ , therefore the system of equations has unique solution.
Using cross multiplication method, we have
$\frac{x}{b_{1} c_{2} - b_{2} c_{1}} = \frac{y}{c_{1} a_{2} - c_{2} a_{1}} = \frac{1}{a_{1} b_{2} - a_{2} b_{1}} \Rightarrow \frac{x}{- 3 (- 13) - 1 \times (- 17)} = \frac{y}{- 17 \times 4 - (- 13) \times 2} = \frac{1}{2 \times 1 - 4 \times (- 3)} \Rightarrow \frac{x}{39 + 17} = \frac{y}{- 68 + 26} = \frac{1}{2 + 12} \Rightarrow \frac{x}{56} = \frac{y}{- 42} = \frac{1}{14}$
$\Rightarrow x = \frac{56}{14}, y = \frac{- 42}{14} \Rightarrow x = 4, y = - 3$
Hence, $x = 4 and y = - 3$ .

Question 3:

The system of equations can be written as
$2 x - 3 y - 17 = 0 . . . . . (i) 4 x + y - 13 = 0 . . . . . (i i)$
The given equations are of the form
$a_{1} x + b_{1} y + c_{1} = 0 and a_{2} x + b_{2} y + c_{2} = 0$
where $a_{1} = 2, b_{1} = - 3, c_{1} = - 17 and a_{2} = 4, b_{2} = 1, c_{2} = - 13$
Now,
$\frac{a_{1}}{a_{2}} = \frac{2}{4} = \frac{1}{2} and \frac{b_{1}}{b_{2}} = \frac{- 3}{1} = - 3$
Since, $\frac{a_{1}}{a_{2}} \neq \frac{b_{1}}{b_{2}}$ , therefore the system of equations has unique solution.
Using cross multiplication method, we have
$\frac{x}{b_{1} c_{2} - b_{2} c_{1}} = \frac{y}{c_{1} a_{2} - c_{2} a_{1}} = \frac{1}{a_{1} b_{2} - a_{2} b_{1}} \Rightarrow \frac{x}{- 3 (- 13) - 1 \times (- 17)} = \frac{y}{- 17 \times 4 - (- 13) \times 2} = \frac{1}{2 \times 1 - 4 \times (- 3)} \Rightarrow \frac{x}{39 + 17} = \frac{y}{- 68 + 26} = \frac{1}{2 + 12} \Rightarrow \frac{x}{56} = \frac{y}{- 42} = \frac{1}{14}$
$\Rightarrow x = \frac{56}{14}, y = \frac{- 42}{14} \Rightarrow x = 4, y = - 3$
Hence, $x = 4 and y = - 3$ .

Answer:

The given system of equations are:
$\frac{x}{3} + \frac{y}{2} = 3$
⇒ $\frac{2 x + 3 y}{6} = 3$
2x + 3y = 18
⇒ 2x + 3y − 18 = 0                 ...(i)
and
x − 2y = 2
x − 2y − 2 = 0                    ...(ii)
These equations are of the forms:
a₁x + b₁y + c₁= 0 and a₂x + b₂y + c₂ = 0
where, a₁ = 2, b₁= 3, c₁ = −18 and a₂ = 1, b₂= −2, c₂ = −2
For a unique solution, we must have:
$\frac{a_{1}}{a_{2}} \neq \frac{b_{1}}{b_{2}}$ , i.e., $\frac{2}{1} \neq \frac{3}{- 2}$
Hence, the given system of equations has a unique solution.

Again, the given equations are:
2x + 3y − 18 = 0           ...(iii)
x − 2y − 2 = 0               ...(iv)
On multiplying (i) by 2 and (ii) by 3, we get:
4x + 6y − 36 = 0           ...(v)
3x − 6y − 6 = 0             ...(vi)
On adding (v) and (vi), we get:
7x = 42
⇒ x = 6
On substituting x = 6 in (iii), we get:
2(6) + 3y = 18
⇒ 3y = (18 − 12) = 6
⇒ y = 2
Hence, x = 6 and y = 2 is the required solution.

Question 4:

The given system of equations are:
$\frac{x}{3} + \frac{y}{2} = 3$
⇒ $\frac{2 x + 3 y}{6} = 3$
2x + 3y = 18
⇒ 2x + 3y − 18 = 0                 ...(i)
and
x − 2y = 2
x − 2y − 2 = 0                    ...(ii)
These equations are of the forms:
a₁x + b₁y + c₁= 0 and a₂x + b₂y + c₂ = 0
where, a₁ = 2, b₁= 3, c₁ = −18 and a₂ = 1, b₂= −2, c₂ = −2
For a unique solution, we must have:
$\frac{a_{1}}{a_{2}} \neq \frac{b_{1}}{b_{2}}$ , i.e., $\frac{2}{1} \neq \frac{3}{- 2}$
Hence, the given system of equations has a unique solution.

Again, the given equations are:
2x + 3y − 18 = 0           ...(iii)
x − 2y − 2 = 0               ...(iv)
On multiplying (i) by 2 and (ii) by 3, we get:
4x + 6y − 36 = 0           ...(v)
3x − 6y − 6 = 0             ...(vi)
On adding (v) and (vi), we get:
7x = 42
⇒ x = 6
On substituting x = 6 in (iii), we get:
2(6) + 3y = 18
⇒ 3y = (18 − 12) = 6
⇒ y = 2
Hence, x = 6 and y = 2 is the required solution.

Answer:

The given system of equations are
$2 x + 3 y - 5 = 0 . . . . . (i) k x - 6 y - 8 = 0 . . . . . (i i)$
This system is of the form
$a_{1} x + b_{1} y + c_{1} = 0 and a_{2} x + b_{2} y + c_{2} = 0$
where $a_{1} = 2, b_{1} = 3, c_{1} = - 5 and a_{2} = k, b_{2} = - 6, c_{2} = - 8$
Now, for the given system of equations to have a unique solution, we must have
$\frac{a_{1}}{a_{2}} \neq \frac{b_{1}}{b_{2}} \Rightarrow \frac{2}{k} \neq \frac{3}{- 6} \Rightarrow k \neq - 4$
Hence, $k \neq - 4$ .

Question 5:

The given system of equations are
$2 x + 3 y - 5 = 0 . . . . . (i) k x - 6 y - 8 = 0 . . . . . (i i)$
This system is of the form
$a_{1} x + b_{1} y + c_{1} = 0 and a_{2} x + b_{2} y + c_{2} = 0$
where $a_{1} = 2, b_{1} = 3, c_{1} = - 5 and a_{2} = k, b_{2} = - 6, c_{2} = - 8$
Now, for the given system of equations to have a unique solution, we must have
$\frac{a_{1}}{a_{2}} \neq \frac{b_{1}}{b_{2}} \Rightarrow \frac{2}{k} \neq \frac{3}{- 6} \Rightarrow k \neq - 4$
Hence, $k \neq - 4$ .

Answer:

The given system of equations are
$x - k y - 2 = 0 . . . . . (i) 3 x + 2 y + 5 = 0 . . . . . (i i)$
This system of equations is of the form
$a_{1} x + b_{1} y + c_{1} = 0 and a_{2} x + b_{2} y + c_{2} = 0$
where $a_{1} = 1, b_{1} = - k, c_{1} = - 2 and a_{2} = 3, b_{2} = 2, c_{2} = 5$
Now, for the given system of equations to have a unique solution, we must have
$\frac{a_{1}}{a_{2}} \neq \frac{b_{1}}{b_{2}} \Rightarrow \frac{1}{3} \neq \frac{- k}{2} \Rightarrow k \neq - \frac{2}{3}$
Hence, $k \neq - \frac{2}{3}$ .

Question 6:

The given system of equations are
$x - k y - 2 = 0 . . . . . (i) 3 x + 2 y + 5 = 0 . . . . . (i i)$
This system of equations is of the form
$a_{1} x + b_{1} y + c_{1} = 0 and a_{2} x + b_{2} y + c_{2} = 0$
where $a_{1} = 1, b_{1} = - k, c_{1} = - 2 and a_{2} = 3, b_{2} = 2, c_{2} = 5$
Now, for the given system of equations to have a unique solution, we must have
$\frac{a_{1}}{a_{2}} \neq \frac{b_{1}}{b_{2}} \Rightarrow \frac{1}{3} \neq \frac{- k}{2} \Rightarrow k \neq - \frac{2}{3}$
Hence, $k \neq - \frac{2}{3}$ .

Answer:

The given system of equations is
$5 x - 7 y - 5 = 0 . . . . . (i) 2 x + k y - 1 = 0 . . . . . (i i)$
This system is of the form
$a_{1} x + b_{1} y + c_{1} = 0 a_{2} x + b_{2} y + c_{2} = 0$
where $a_{1} = 5, b_{1} = - 7, c_{1} = - 5 and a_{2} = 2, b_{2} = k, c_{2} = - 1$
For the given system of equations to have a unique solution, we must have
$\frac{a_{1}}{a_{2}} \neq \frac{b_{1}}{b_{2}} \Rightarrow \frac{5}{2} \neq \frac{- 7}{k} \Rightarrow k \neq \frac{- 14}{5}$
Hence, $k \neq - \frac{14}{5}$ .

Question 7:

The given system of equations is
$5 x - 7 y - 5 = 0 . . . . . (i) 2 x + k y - 1 = 0 . . . . . (i i)$
This system is of the form
$a_{1} x + b_{1} y + c_{1} = 0 a_{2} x + b_{2} y + c_{2} = 0$
where $a_{1} = 5, b_{1} = - 7, c_{1} = - 5 and a_{2} = 2, b_{2} = k, c_{2} = - 1$
For the given system of equations to have a unique solution, we must have
$\frac{a_{1}}{a_{2}} \neq \frac{b_{1}}{b_{2}} \Rightarrow \frac{5}{2} \neq \frac{- 7}{k} \Rightarrow k \neq \frac{- 14}{5}$
Hence, $k \neq - \frac{14}{5}$ .

Answer:

The given system of equations is
$4 x + k y + 8 = 0 . . . . . (i) x + y + 1 = 0 . . . . . (i i)$
This system is of the form
$a_{1} x + b_{1} y + c_{1} = 0 a_{2} x + b_{2} y + c_{2} = 0$
where $a_{1} = 4, b_{1} = k, c_{1} = 8 and a_{2} = 1, b_{2} = 1, c_{2} = 1$
For the given system of equations to have a unique solution, we must have
$\frac{a_{1}}{a_{2}} \neq \frac{b_{1}}{b_{2}} \Rightarrow \frac{4}{1} \neq \frac{k}{1} \Rightarrow k \neq 4$
Hence, $k \neq 4$ .

Question 8:

The given system of equations is
$4 x + k y + 8 = 0 . . . . . (i) x + y + 1 = 0 . . . . . (i i)$
This system is of the form
$a_{1} x + b_{1} y + c_{1} = 0 a_{2} x + b_{2} y + c_{2} = 0$
where $a_{1} = 4, b_{1} = k, c_{1} = 8 and a_{2} = 1, b_{2} = 1, c_{2} = 1$
For the given system of equations to have a unique solution, we must have
$\frac{a_{1}}{a_{2}} \neq \frac{b_{1}}{b_{2}} \Rightarrow \frac{4}{1} \neq \frac{k}{1} \Rightarrow k \neq 4$
Hence, $k \neq 4$ .

Answer:

The given system of equations:
4x − 5y = k
⇒ 4x − 5y − k = 0 ...(i)
And, 2x − 3y = 12
⇒ 2x − 3y − 12 = 0 ...(ii)
These equations are of the following form:
a₁x + b₁y + c₁= 0, a₂x + b₂y + c₂ = 0
Here, a₁ = 4, b₁= −5, c₁ = −k and a₂ = 2, b₂= −3, c₂ = −12
For a unique solution, we must have:
$\frac{a_{1}}{a_{2}} \neq \frac{b_{1}}{b_{2}}$
i.e. $\frac{4}{2} \neq \frac{- 5}{- 3}$
$\Rightarrow 2 \neq \frac{5}{3} \Rightarrow 6 \neq 5$
Thus, for all real values of k, the given system of equations will have a unique solution.

Question 9:

The given system of equations:
4x − 5y = k
⇒ 4x − 5y − k = 0 ...(i)
And, 2x − 3y = 12
⇒ 2x − 3y − 12 = 0 ...(ii)
These equations are of the following form:
a₁x + b₁y + c₁= 0, a₂x + b₂y + c₂ = 0
Here, a₁ = 4, b₁= −5, c₁ = −k and a₂ = 2, b₂= −3, c₂ = −12
For a unique solution, we must have:
$\frac{a_{1}}{a_{2}} \neq \frac{b_{1}}{b_{2}}$
i.e. $\frac{4}{2} \neq \frac{- 5}{- 3}$
$\Rightarrow 2 \neq \frac{5}{3} \Rightarrow 6 \neq 5$
Thus, for all real values of k, the given system of equations will have a unique solution.

Answer:

The given system of equations:
kx + 3y = (k − 3)
⇒ kx + 3y − (k − 3) = 0 ....(i)
And, 12x + ky = k
⇒ 12x + ky − k = 0 ....(ii)
These equations are of the following form:
a₁x + b₁y + c₁= 0, a₂x + b₂y + c₂ = 0
Here, a₁ = k, b₁= 3, c₁ = −(k − 3) and a₂ = 12, b₂= k, c₂ = −k
For a unique solution, we must have:
$\frac{a_{1}}{a_{2}} \neq \frac{b_{1}}{b_{2}}$
i.e. $\frac{k}{12} \neq \frac{3}{k}$
$\Rightarrow k^{2} \neq 36 \Rightarrow k \neq \pm 6$
Thus, for all real values of k other than $\pm 6$ , the given system of equations will have a unique solution.

Question 10:

The given system of equations:
kx + 3y = (k − 3)
⇒ kx + 3y − (k − 3) = 0 ....(i)
And, 12x + ky = k
⇒ 12x + ky − k = 0 ....(ii)
These equations are of the following form:
a₁x + b₁y + c₁= 0, a₂x + b₂y + c₂ = 0
Here, a₁ = k, b₁= 3, c₁ = −(k − 3) and a₂ = 12, b₂= k, c₂ = −k
For a unique solution, we must have:
$\frac{a_{1}}{a_{2}} \neq \frac{b_{1}}{b_{2}}$
i.e. $\frac{k}{12} \neq \frac{3}{k}$
$\Rightarrow k^{2} \neq 36 \Rightarrow k \neq \pm 6$
Thus, for all real values of k other than $\pm 6$ , the given system of equations will have a unique solution.

Answer:

The given system of equations is:
2x − 3y = 5
⇒ 2x − 3y − 5 = 0 ...(i)
6x − 9y = 15
⇒ 6x − 9y − 15 = 0 ...(ii)
These equations are of the forms:
a₁x + b₁y + c₁= 0 and a₂x + b₂y + c₂ = 0
where, a₁ = 2, b₁= −3, c₁ = −5 and a₂ = 6, b₂= −9, c₂ = −15

$∴ \frac{a_{1}}{a_{2}} = \frac{2}{6} = \frac{1}{3}, \frac{b_{1}}{b 2} = \frac{- 3}{- 9} = \frac{1}{3}$ and $\frac{c_{1}}{c_{2}} = \frac{- 5}{- 15} = \frac{1}{3}$
Thus, $\frac{a_{1}}{a_{2}} = \frac{b_{1}}{b_{2}} = \frac{c_{1}}{c_{2}}$
Hence, the given system of equations has an infinite number of solutions.

Question 11:

The given system of equations is:
2x − 3y = 5
⇒ 2x − 3y − 5 = 0 ...(i)
6x − 9y = 15
⇒ 6x − 9y − 15 = 0 ...(ii)
These equations are of the forms:
a₁x + b₁y + c₁= 0 and a₂x + b₂y + c₂ = 0
where, a₁ = 2, b₁= −3, c₁ = −5 and a₂ = 6, b₂= −9, c₂ = −15

$∴ \frac{a_{1}}{a_{2}} = \frac{2}{6} = \frac{1}{3}, \frac{b_{1}}{b 2} = \frac{- 3}{- 9} = \frac{1}{3}$ and $\frac{c_{1}}{c_{2}} = \frac{- 5}{- 15} = \frac{1}{3}$
Thus, $\frac{a_{1}}{a_{2}} = \frac{b_{1}}{b_{2}} = \frac{c_{1}}{c_{2}}$
Hence, the given system of equations has an infinite number of solutions.

Answer:

The given system of equations can be written as
$6 x + 5 y - 11 = . . . . . (i) 9 x + \frac{15}{2} y - 21 = 0 . . . . . (i i)$
This system is of the form
$a_{1} x + b_{1} y + c_{1} = 0 a_{2} x + b_{2} y + c_{2} = 0$
where $a_{1} = 6, b_{1} = 5, c_{1} = - 11 and a_{2} = 9, b_{2} = \frac{15}{2}, c_{2} = - 21$
Now,
$\frac{a_{1}}{a_{2}} = \frac{6}{9} = \frac{2}{3} \frac{b_{1}}{b_{2}} = \frac{5}{\frac{15}{2}} = \frac{2}{3} \frac{c_{1}}{c_{2}} = \frac{- 11}{- 21} = \frac{11}{21}$
Since, $\frac{a_{1}}{a_{2}} = \frac{b_{1}}{b_{2}} \neq \frac{c_{1}}{c_{2}}$ , therefore the given system has no solution.

Question 12:

The given system of equations can be written as
$6 x + 5 y - 11 = . . . . . (i) 9 x + \frac{15}{2} y - 21 = 0 . . . . . (i i)$
This system is of the form
$a_{1} x + b_{1} y + c_{1} = 0 a_{2} x + b_{2} y + c_{2} = 0$
where $a_{1} = 6, b_{1} = 5, c_{1} = - 11 and a_{2} = 9, b_{2} = \frac{15}{2}, c_{2} = - 21$
Now,
$\frac{a_{1}}{a_{2}} = \frac{6}{9} = \frac{2}{3} \frac{b_{1}}{b_{2}} = \frac{5}{\frac{15}{2}} = \frac{2}{3} \frac{c_{1}}{c_{2}} = \frac{- 11}{- 21} = \frac{11}{21}$
Since, $\frac{a_{1}}{a_{2}} = \frac{b_{1}}{b_{2}} \neq \frac{c_{1}}{c_{2}}$ , therefore the given system has no solution.

Answer:

The given system of equations is:
kx + 2y = 5
⇒ kx + 2y − 5= 0 ...(i)
3x − 4y = 10
⇒ 3x − 4y − 10 = 0 ...(ii)
These equations are of the forms:
a₁x + b₁y + c₁= 0 and a₂x + b₂y + c₂ = 0
where, a₁ = k, b₁= 2, c₁ = −5 and a₂ = 3, b₂= −4, c₂ = −10
(i) For a unique solution, we must have:
$\frac{a_{1}}{a_{2}} \neq \frac{b_{1}}{b_{2}}$ , i.e., $\frac{k}{3} \neq \frac{2}{- 4} \Rightarrow k \neq \frac{- 3}{2}$
Thus for all real values of k other than $\frac{- 3}{2}$ , the given system of equations will have a unique solution.
(ii) For the given system of equations to have no solutions, we must have:
$\frac{a_{1}}{a_{2}} = \frac{b_{1}}{b_{2}} \neq \frac{c_{1}}{c_{2}}$
$\Rightarrow \frac{k}{3} = \frac{2}{- 4} \neq \frac{- 5}{- 10}$
$\Rightarrow \frac{k}{3} = \frac{2}{- 4} and \frac{k}{3} \neq \frac{1}{2}$
$\Rightarrow k = \frac{- 3}{2}, k \neq \frac{3}{2}$
Hence, the required value of k is $\frac{- 3}{2}$ .

Question 13:

The given system of equations is:
kx + 2y = 5
⇒ kx + 2y − 5= 0 ...(i)
3x − 4y = 10
⇒ 3x − 4y − 10 = 0 ...(ii)
These equations are of the forms:
a₁x + b₁y + c₁= 0 and a₂x + b₂y + c₂ = 0
where, a₁ = k, b₁= 2, c₁ = −5 and a₂ = 3, b₂= −4, c₂ = −10
(i) For a unique solution, we must have:
$\frac{a_{1}}{a_{2}} \neq \frac{b_{1}}{b_{2}}$ , i.e., $\frac{k}{3} \neq \frac{2}{- 4} \Rightarrow k \neq \frac{- 3}{2}$
Thus for all real values of k other than $\frac{- 3}{2}$ , the given system of equations will have a unique solution.
(ii) For the given system of equations to have no solutions, we must have:
$\frac{a_{1}}{a_{2}} = \frac{b_{1}}{b_{2}} \neq \frac{c_{1}}{c_{2}}$
$\Rightarrow \frac{k}{3} = \frac{2}{- 4} \neq \frac{- 5}{- 10}$
$\Rightarrow \frac{k}{3} = \frac{2}{- 4} and \frac{k}{3} \neq \frac{1}{2}$
$\Rightarrow k = \frac{- 3}{2}, k \neq \frac{3}{2}$
Hence, the required value of k is $\frac{- 3}{2}$ .

Answer:

The given system of equations is:
x + 2y = 5
⇒ x + 2y − 5= 0 ...(i)
3x + ky + 15 = 0 ...(ii)
These equations are of the form:
a₁x + b₁y + c₁= 0 and a₂x + b₂y + c₂ = 0
where, a₁ = 1, b₁= 2, c₁ = −5 and a₂ = 3, b₂= k, c₂ = 15
(i) For a unique solution, we must have:
$\frac{a_{1}}{a_{2}} \neq \frac{b_{1}}{b_{2}}$ , i.e., $\frac{1}{3} \neq \frac{2}{k} \Rightarrow k \neq 6$
Thus, for all real values of k other than 6, the given system of equations will have a unique solution.
(ii) For the given system of equations to have no solutions, we must have:
$\frac{a_{1}}{a_{2}} = \frac{b_{1}}{b_{2}} \neq \frac{c_{1}}{c_{2}}$
$\Rightarrow \frac{1}{3} = \frac{2}{k} \neq \frac{- 5}{15}$
$\Rightarrow \frac{1}{3} = \frac{2}{k} and \frac{2}{k} \neq \frac{- 5}{15}$
$\Rightarrow k = 6, k \neq - 6$
Hence, the required value of k is 6.

Question 14:

The given system of equations is:
x + 2y = 5
⇒ x + 2y − 5= 0 ...(i)
3x + ky + 15 = 0 ...(ii)
These equations are of the form:
a₁x + b₁y + c₁= 0 and a₂x + b₂y + c₂ = 0
where, a₁ = 1, b₁= 2, c₁ = −5 and a₂ = 3, b₂= k, c₂ = 15
(i) For a unique solution, we must have:
$\frac{a_{1}}{a_{2}} \neq \frac{b_{1}}{b_{2}}$ , i.e., $\frac{1}{3} \neq \frac{2}{k} \Rightarrow k \neq 6$
Thus, for all real values of k other than 6, the given system of equations will have a unique solution.
(ii) For the given system of equations to have no solutions, we must have:
$\frac{a_{1}}{a_{2}} = \frac{b_{1}}{b_{2}} \neq \frac{c_{1}}{c_{2}}$
$\Rightarrow \frac{1}{3} = \frac{2}{k} \neq \frac{- 5}{15}$
$\Rightarrow \frac{1}{3} = \frac{2}{k} and \frac{2}{k} \neq \frac{- 5}{15}$
$\Rightarrow k = 6, k \neq - 6$
Hence, the required value of k is 6.

Answer:

The given system of equations is:
x + 2y = 3
x + 2y − 3= 0 ....(i)
And, 5x + ky + 7 = 0 ....(ii)
These equations are of the following form:
a₁x + b₁y + c₁= 0, a₂x + b₂y + c₂ = 0
Here, a₁ = 1, b₁= 2, c₁ = −3 and a₂ = 5, b₂= k, c₂ = 7
(i) For a unique solution, we must have:
$\frac{a_{1}}{a_{2}} \neq \frac{b_{1}}{b_{2}}$ , i.e. $\frac{1}{5} \neq \frac{2}{k} \Rightarrow k \neq 10$
Thus, for all real values of kâ€‹, other than 10, the given system of equations will have a unique solution.
(ii) In order that the given system of equations has no solution, we must have:
$\frac{a_{1}}{a_{2}} = \frac{b_{1}}{b_{2}} \neq \frac{c_{1}}{c_{2}}$
$\frac{1}{5} = \frac{2}{k} \neq \frac{- 3}{7}$
$\Rightarrow \frac{1}{5} = \frac{2}{k} and \frac{2}{k} \neq \frac{- 3}{7}$
$\Rightarrow k = 10, k \neq \frac{14}{- 3}$
Hence, the required value of k is 10.
There is no value of k for which the given system of equations has an infinite number of solutions.

Question 15:

The given system of equations is:
x + 2y = 3
x + 2y − 3= 0 ....(i)
And, 5x + ky + 7 = 0 ....(ii)
These equations are of the following form:
a₁x + b₁y + c₁= 0, a₂x + b₂y + c₂ = 0
Here, a₁ = 1, b₁= 2, c₁ = −3 and a₂ = 5, b₂= k, c₂ = 7
(i) For a unique solution, we must have:
$\frac{a_{1}}{a_{2}} \neq \frac{b_{1}}{b_{2}}$ , i.e. $\frac{1}{5} \neq \frac{2}{k} \Rightarrow k \neq 10$
Thus, for all real values of kâ€‹, other than 10, the given system of equations will have a unique solution.
(ii) In order that the given system of equations has no solution, we must have:
$\frac{a_{1}}{a_{2}} = \frac{b_{1}}{b_{2}} \neq \frac{c_{1}}{c_{2}}$
$\frac{1}{5} = \frac{2}{k} \neq \frac{- 3}{7}$
$\Rightarrow \frac{1}{5} = \frac{2}{k} and \frac{2}{k} \neq \frac{- 3}{7}$
$\Rightarrow k = 10, k \neq \frac{14}{- 3}$
Hence, the required value of k is 10.
There is no value of k for which the given system of equations has an infinite number of solutions.

Answer:

The given system of equations:
2x + 3y = 7
⇒ 2x + 3y − 7 = 0 ....(i)
And, (k − 1)x + (k + 2)y = 3k
⇒ (k − 1)x + (k + 2)y − 3k = 0 ....(ii)
These equations are of the following form:
a₁x + b₁y + c₁= 0, a₂x + b₂y + c₂ = 0
Here, a₁ = 2, b₁= 3, c₁ = −7 and a₂ = (k − 1), b₂= (k + 2), c₂ = −3k
For an infinite number of solutions, we must have:
$\frac{a_{1}}{a_{2}} = \frac{b_{1}}{b_{2}} = \frac{c_{1}}{c_{2}}$
$\frac{2}{(k - 1)} = \frac{3}{(k + 2)} = \frac{- 7}{- 3 k}$
$\Rightarrow \frac{2}{(k - 1)} = \frac{3}{(k + 2)} = \frac{7}{3 k}$

Now, we have the following three cases:
Case I:
$\frac{2}{k - 1} = \frac{3}{k + 2}$
$\Rightarrow 2 (k + 2) = 3 (k - 1) \Rightarrow 2 k + 4 = 3 k - 3 \Rightarrow k = 7$

Case II:
$\frac{3}{k + 2} = \frac{7}{3 k}$
$\Rightarrow 7 (k + 2) = 9 k \Rightarrow 7 k + 14 = 9 k \Rightarrow 2 k = 14 \Rightarrow k = 7$

Case III:
$\frac{2}{k - 1} = \frac{7}{3 k}$
$\Rightarrow 7 k - 7 = 6 k \Rightarrow k = 7$

Hence, the given system of equations has an infinite number of solutions when k is equal to 7.

Question 16:

The given system of equations:
2x + 3y = 7
⇒ 2x + 3y − 7 = 0 ....(i)
And, (k − 1)x + (k + 2)y = 3k
⇒ (k − 1)x + (k + 2)y − 3k = 0 ....(ii)
These equations are of the following form:
a₁x + b₁y + c₁= 0, a₂x + b₂y + c₂ = 0
Here, a₁ = 2, b₁= 3, c₁ = −7 and a₂ = (k − 1), b₂= (k + 2), c₂ = −3k
For an infinite number of solutions, we must have:
$\frac{a_{1}}{a_{2}} = \frac{b_{1}}{b_{2}} = \frac{c_{1}}{c_{2}}$
$\frac{2}{(k - 1)} = \frac{3}{(k + 2)} = \frac{- 7}{- 3 k}$
$\Rightarrow \frac{2}{(k - 1)} = \frac{3}{(k + 2)} = \frac{7}{3 k}$

Now, we have the following three cases:
Case I:
$\frac{2}{k - 1} = \frac{3}{k + 2}$
$\Rightarrow 2 (k + 2) = 3 (k - 1) \Rightarrow 2 k + 4 = 3 k - 3 \Rightarrow k = 7$

Case II:
$\frac{3}{k + 2} = \frac{7}{3 k}$
$\Rightarrow 7 (k + 2) = 9 k \Rightarrow 7 k + 14 = 9 k \Rightarrow 2 k = 14 \Rightarrow k = 7$

Case III:
$\frac{2}{k - 1} = \frac{7}{3 k}$
$\Rightarrow 7 k - 7 = 6 k \Rightarrow k = 7$

Hence, the given system of equations has an infinite number of solutions when k is equal to 7.

Answer:

The given system of equations:
2x + (k − 2)y = k
⇒ 2x + (k − 2)y − k = 0 ....(i)
And, 6x + (2k − 1)y = (2k + 5)
⇒ 6x + (2k − 1)y − (2k + 5) = 0 ....(ii)
These equations are of the following form:
a₁x + b₁y + c₁= 0, a₂x + b₂y + c₂ = 0
Here, a₁ = 2, b₁= (k − 2), c₁ = −k and a₂ = 6, b₂= (2k − 1), c₂ = −(2k + 5)
For an infinite number of solutions, we must have:
$\frac{a_{1}}{a_{2}} = \frac{b_{1}}{b_{2}} = \frac{c_{1}}{c_{2}}$
$\frac{2}{6} = \frac{(k - 2)}{(2 k - 1)} = \frac{- k}{- (2 k + 5)}$
$\Rightarrow \frac{1}{3} = \frac{(k - 2)}{(2 k - 1)} = \frac{k}{(2 k + 5)}$

Now, we have the following three cases:
Case I:
$\frac{1}{3} = \frac{k - 2}{2 k - 1}$
$\Rightarrow (2 k - 1) = 3 (k - 2)$
$\Rightarrow 2 k - 1 = 3 k - 6 \Rightarrow k = 5$

Case II:
$\frac{k - 2}{2 k - 1} = \frac{k}{2 k + 5}$
$\Rightarrow (k - 2) (2 k + 5) = k (2 k - 1)$
$\Rightarrow 2 k^{2} + 5 k - 4 k - 10 = 2 k^{2} - k$
$\Rightarrow k + k = 10 \Rightarrow 2 k = 10 \Rightarrow k = 5$

Case III:
$\frac{1}{3} = \frac{k}{2 k + 5}$
$\Rightarrow 2 k + 5 = 3 k \Rightarrow k = 5$

Hence, the given system of equations has an infinite number of solutions when k is equal to 5.

Question 17:

The given system of equations:
2x + (k − 2)y = k
⇒ 2x + (k − 2)y − k = 0 ....(i)
And, 6x + (2k − 1)y = (2k + 5)
⇒ 6x + (2k − 1)y − (2k + 5) = 0 ....(ii)
These equations are of the following form:
a₁x + b₁y + c₁= 0, a₂x + b₂y + c₂ = 0
Here, a₁ = 2, b₁= (k − 2), c₁ = −k and a₂ = 6, b₂= (2k − 1), c₂ = −(2k + 5)
For an infinite number of solutions, we must have:
$\frac{a_{1}}{a_{2}} = \frac{b_{1}}{b_{2}} = \frac{c_{1}}{c_{2}}$
$\frac{2}{6} = \frac{(k - 2)}{(2 k - 1)} = \frac{- k}{- (2 k + 5)}$
$\Rightarrow \frac{1}{3} = \frac{(k - 2)}{(2 k - 1)} = \frac{k}{(2 k + 5)}$

Now, we have the following three cases:
Case I:
$\frac{1}{3} = \frac{k - 2}{2 k - 1}$
$\Rightarrow (2 k - 1) = 3 (k - 2)$
$\Rightarrow 2 k - 1 = 3 k - 6 \Rightarrow k = 5$

Case II:
$\frac{k - 2}{2 k - 1} = \frac{k}{2 k + 5}$
$\Rightarrow (k - 2) (2 k + 5) = k (2 k - 1)$
$\Rightarrow 2 k^{2} + 5 k - 4 k - 10 = 2 k^{2} - k$
$\Rightarrow k + k = 10 \Rightarrow 2 k = 10 \Rightarrow k = 5$

Case III:
$\frac{1}{3} = \frac{k}{2 k + 5}$
$\Rightarrow 2 k + 5 = 3 k \Rightarrow k = 5$

Hence, the given system of equations has an infinite number of solutions when k is equal to 5.

Answer:

The given system of equations:
kx + 3y = (2k + 1)
⇒ kx + 3y − (2k + 1) = 0 ...(i)
And, 2(k + 1)x + 9y = (7k + 1)
⇒ 2(k + 1)x + 9y − (7k + 1) = 0 ...(ii)
These equations are of the following form:
a₁x + b₁y + c₁= 0, a₂x + b₂y + c₂ = 0
Here, a₁ = k, b₁= 3, c₁ = −(2k + 1) and a₂ = 2(k + 1), b₂= 9, c₂ = −(7k + 1)
For an infinite number of solutions, we must have:
$\frac{a_{1}}{a_{2}} = \frac{b_{1}}{b_{2}} = \frac{c_{1}}{c_{2}}$
$i . e . \frac{k}{2 (k + 1)} = \frac{3}{9} = \frac{- (2 k + 1)}{- (7 k + 1)}$
$\Rightarrow \frac{k}{2 (k + 1)} = \frac{1}{3} = \frac{(2 k + 1)}{(7 k + 1)}$

Now, we have the following three cases:
Case I:
$\frac{k}{2 (k + 1)} = \frac{1}{3}$
$\Rightarrow 2 (k + 1) = 3 k$
$\Rightarrow 2 k + 2 = 3 k$
$\Rightarrow k = 2$

Case II:
$\frac{1}{3} = \frac{2 k + 1}{7 k + 1}$
$\Rightarrow (7 k + 1) = 6 k + 3$
$\Rightarrow k = 2$

Case III:
$\frac{k}{2 (k + 1)} = \frac{2 k + 1}{7 k + 1}$
$\Rightarrow k (7 k + 1) = (2 k + 1) \times 2 (k + 1)$
$\Rightarrow 7 k^{2} + k = (2 k + 1) (2 k + 2)$
$\Rightarrow 7 k^{2} + k = 4 k^{2} + 4 k + 2 k + 2$
$\Rightarrow 3 k^{2} - 5 k - 2 = 0$
$\Rightarrow 3 k^{2} - 6 k + k - 2 = 0$
$\Rightarrow 3 k (k - 2) + 1 (k - 2) = 0$
$\Rightarrow (3 k + 1) (k - 2) = 0$
$\Rightarrow k = 2 or k = \frac{- 1}{3}$
Hence, the given system of equations has an infinite number of solutions when k is equal to 2.

Question 18:

The given system of equations:
kx + 3y = (2k + 1)
⇒ kx + 3y − (2k + 1) = 0 ...(i)
And, 2(k + 1)x + 9y = (7k + 1)
⇒ 2(k + 1)x + 9y − (7k + 1) = 0 ...(ii)
These equations are of the following form:
a₁x + b₁y + c₁= 0, a₂x + b₂y + c₂ = 0
Here, a₁ = k, b₁= 3, c₁ = −(2k + 1) and a₂ = 2(k + 1), b₂= 9, c₂ = −(7k + 1)
For an infinite number of solutions, we must have:
$\frac{a_{1}}{a_{2}} = \frac{b_{1}}{b_{2}} = \frac{c_{1}}{c_{2}}$
$i . e . \frac{k}{2 (k + 1)} = \frac{3}{9} = \frac{- (2 k + 1)}{- (7 k + 1)}$
$\Rightarrow \frac{k}{2 (k + 1)} = \frac{1}{3} = \frac{(2 k + 1)}{(7 k + 1)}$

Now, we have the following three cases:
Case I:
$\frac{k}{2 (k + 1)} = \frac{1}{3}$
$\Rightarrow 2 (k + 1) = 3 k$
$\Rightarrow 2 k + 2 = 3 k$
$\Rightarrow k = 2$

Case II:
$\frac{1}{3} = \frac{2 k + 1}{7 k + 1}$
$\Rightarrow (7 k + 1) = 6 k + 3$
$\Rightarrow k = 2$

Case III:
$\frac{k}{2 (k + 1)} = \frac{2 k + 1}{7 k + 1}$
$\Rightarrow k (7 k + 1) = (2 k + 1) \times 2 (k + 1)$
$\Rightarrow 7 k^{2} + k = (2 k + 1) (2 k + 2)$
$\Rightarrow 7 k^{2} + k = 4 k^{2} + 4 k + 2 k + 2$
$\Rightarrow 3 k^{2} - 5 k - 2 = 0$
$\Rightarrow 3 k^{2} - 6 k + k - 2 = 0$
$\Rightarrow 3 k (k - 2) + 1 (k - 2) = 0$
$\Rightarrow (3 k + 1) (k - 2) = 0$
$\Rightarrow k = 2 or k = \frac{- 1}{3}$
Hence, the given system of equations has an infinite number of solutions when k is equal to 2.

Answer:

The given system of equations:
5x + 2y = 2k
⇒ 5x + 2y − 2k= 0 ...(i)
And, 2(k + 1)x + ky = (3k + 4)
⇒ 2(k + 1)x + ky − (3k + 4) = 0 ...(ii)
These equations are of the following form:
a₁x + b₁y + c₁= 0, a₂x + b₂y + c₂ = 0
Here, a₁ = 5, b₁= 2, c₁ = −2k and a₂ = 2(k + 1), b₂= k, c₂ = −(3k + 4)
For an infinite number of solutions, we must have:
$\frac{a_{1}}{a_{2}} = \frac{b_{1}}{b_{2}} = \frac{c_{1}}{c_{2}}$
$\frac{5}{2 (k + 1)} = \frac{2}{k} = \frac{- 2 k}{- (3 k + 4)}$
$\Rightarrow \frac{5}{2 (k + 1)} = \frac{2}{k} = \frac{2 k}{(3 k + 4)}$

Now, we have the following three cases:
Case I:
$\frac{5}{2 (k + 1)} = \frac{2}{k}$
$\Rightarrow 2 \times 2 (k + 1) = 5 k \Rightarrow 4 (k + 1) = 5 k$
$\Rightarrow 4 k + 4 = 5 k \Rightarrow k = 4$

Case II:
$\frac{2}{k} = \frac{2 k}{(3 k + 4)}$
$\Rightarrow 2 k^{2} = 2 \times (3 k + 4)$
$\Rightarrow 2 k^{2} = 6 k + 8 \Rightarrow 2 k^{2} - 6 k - 8 = 0$
$\Rightarrow 2 (k^{2} - 3 k - 4) = 0$
$\Rightarrow k^{2} - 4 k + k - 4 = 0$
$\Rightarrow k (k - 4) + 1 (k - 4) = 0$
$\Rightarrow (k + 1) (k - 4) = 0$
$\Rightarrow (k + 1) = 0 or (k - 4) = 0$
$\Rightarrow k = - 1 or k = 4$

Case III:
$\frac{5}{2 (k + 1)} = \frac{2 k}{3 k + 4}$
$\Rightarrow 15 k + 20 = 4 k^{2} + 4 k$
$\Rightarrow 4 k^{2} - 11 k - 20 = 0$
$\Rightarrow 4 k^{2} - 16 k + 5 k - 20 = 0$
$\Rightarrow 4 k (k - 4) + 5 (k - 4) = 0$
$\Rightarrow (k - 4) (4 k + 5) = 0$
$\Rightarrow k = 4 or k = \frac{- 5}{4}$

Hence, the given system of equations has an infinite number of solutions when k is equal to 4.

Question 19:

The given system of equations:
5x + 2y = 2k
⇒ 5x + 2y − 2k= 0 ...(i)
And, 2(k + 1)x + ky = (3k + 4)
⇒ 2(k + 1)x + ky − (3k + 4) = 0 ...(ii)
These equations are of the following form:
a₁x + b₁y + c₁= 0, a₂x + b₂y + c₂ = 0
Here, a₁ = 5, b₁= 2, c₁ = −2k and a₂ = 2(k + 1), b₂= k, c₂ = −(3k + 4)
For an infinite number of solutions, we must have:
$\frac{a_{1}}{a_{2}} = \frac{b_{1}}{b_{2}} = \frac{c_{1}}{c_{2}}$
$\frac{5}{2 (k + 1)} = \frac{2}{k} = \frac{- 2 k}{- (3 k + 4)}$
$\Rightarrow \frac{5}{2 (k + 1)} = \frac{2}{k} = \frac{2 k}{(3 k + 4)}$

Now, we have the following three cases:
Case I:
$\frac{5}{2 (k + 1)} = \frac{2}{k}$
$\Rightarrow 2 \times 2 (k + 1) = 5 k \Rightarrow 4 (k + 1) = 5 k$
$\Rightarrow 4 k + 4 = 5 k \Rightarrow k = 4$

Case II:
$\frac{2}{k} = \frac{2 k}{(3 k + 4)}$
$\Rightarrow 2 k^{2} = 2 \times (3 k + 4)$
$\Rightarrow 2 k^{2} = 6 k + 8 \Rightarrow 2 k^{2} - 6 k - 8 = 0$
$\Rightarrow 2 (k^{2} - 3 k - 4) = 0$
$\Rightarrow k^{2} - 4 k + k - 4 = 0$
$\Rightarrow k (k - 4) + 1 (k - 4) = 0$
$\Rightarrow (k + 1) (k - 4) = 0$
$\Rightarrow (k + 1) = 0 or (k - 4) = 0$
$\Rightarrow k = - 1 or k = 4$

Case III:
$\frac{5}{2 (k + 1)} = \frac{2 k}{3 k + 4}$
$\Rightarrow 15 k + 20 = 4 k^{2} + 4 k$
$\Rightarrow 4 k^{2} - 11 k - 20 = 0$
$\Rightarrow 4 k^{2} - 16 k + 5 k - 20 = 0$
$\Rightarrow 4 k (k - 4) + 5 (k - 4) = 0$
$\Rightarrow (k - 4) (4 k + 5) = 0$
$\Rightarrow k = 4 or k = \frac{- 5}{4}$

Hence, the given system of equations has an infinite number of solutions when k is equal to 4.

Answer:

The given system of equations:
(k − 1)x − y = 5
⇒ (k − 1)x − y − 5 = 0 ...(i)
And, (k + 1)x + (1 − k)y = (3k + 1)
⇒ (k + 1)x + (1 − k)y − (3k + 1) = 0 ...(ii)
These equations are of the following form:
a₁x + b₁y + c₁= 0, a₂x + b₂y + c₂ = 0
Here, a₁ = (k − 1), b₁= −1, c₁ = −5 and a₂ = (k + 1), b₂= (1 − k), c₂ = −(3k + 1)
For an infinite number of solutions, we must have:
$\frac{a_{1}}{a_{2}} = \frac{b_{1}}{b_{2}} = \frac{c_{1}}{c_{2}}$
$i . e . \frac{(k - 1)}{(k + 1)} = \frac{- 1}{- (k - 1)} = \frac{- 5}{- (3 k + 1)}$
$\Rightarrow \frac{(k - 1)}{(k + 1)} = \frac{1}{(k - 1)} = \frac{5}{(3 k + 1)}$

Now, we have the following three cases:
Case I:
$\frac{(k - 1)}{(k + 1)} = \frac{1}{(k - 1)}$
$\Rightarrow {(k - 1)}^{2} = (k + 1)$
$\Rightarrow k^{2} + 1 - 2 k = k + 1$
$\Rightarrow k^{2} - 3 k = 0 \Rightarrow k (k - 3) = 0 \Rightarrow k = 0 or k = 3$

Case II:
$\frac{1}{(k - 1)} = \frac{5}{(3 k + 1)}$
$\Rightarrow 3 k + 1 = 5 (k - 1)$
$\Rightarrow 3 k + 1 = 5 k - 5$
$\Rightarrow 2 k = 6 \Rightarrow k = 3$

Case III:
$\frac{(k - 1)}{(k + 1)} = \frac{5}{(3 k + 1)}$
$\Rightarrow (3 k + 1) (k - 1) = 5 (k + 1)$
$\Rightarrow 3 k^{2} + k - 3 k - 1 = 5 k + 5$
$\Rightarrow 3 k^{2} - 2 k - 5 k - 1 - 5 = 0$
$\Rightarrow 3 k^{2} - 7 k - 6 = 0$
$\Rightarrow 3 k^{2} - 9 k + 2 k - 6 = 0$
$\Rightarrow 3 k (k - 3) + 2 (k - 3) = 0$
$\Rightarrow (k - 3) (3 k + 2) = 0$
$\Rightarrow (k - 3) = 0 or (3 k + 2) = 0$
$\Rightarrow k = 3 or k = \frac{- 2}{3}$

Hence, the given system of equations has an infinite number of solutions when k is equal to 3.

Question 20:

The given system of equations:
(k − 1)x − y = 5
⇒ (k − 1)x − y − 5 = 0 ...(i)
And, (k + 1)x + (1 − k)y = (3k + 1)
⇒ (k + 1)x + (1 − k)y − (3k + 1) = 0 ...(ii)
These equations are of the following form:
a₁x + b₁y + c₁= 0, a₂x + b₂y + c₂ = 0
Here, a₁ = (k − 1), b₁= −1, c₁ = −5 and a₂ = (k + 1), b₂= (1 − k), c₂ = −(3k + 1)
For an infinite number of solutions, we must have:
$\frac{a_{1}}{a_{2}} = \frac{b_{1}}{b_{2}} = \frac{c_{1}}{c_{2}}$
$i . e . \frac{(k - 1)}{(k + 1)} = \frac{- 1}{- (k - 1)} = \frac{- 5}{- (3 k + 1)}$
$\Rightarrow \frac{(k - 1)}{(k + 1)} = \frac{1}{(k - 1)} = \frac{5}{(3 k + 1)}$

Now, we have the following three cases:
Case I:
$\frac{(k - 1)}{(k + 1)} = \frac{1}{(k - 1)}$
$\Rightarrow {(k - 1)}^{2} = (k + 1)$
$\Rightarrow k^{2} + 1 - 2 k = k + 1$
$\Rightarrow k^{2} - 3 k = 0 \Rightarrow k (k - 3) = 0 \Rightarrow k = 0 or k = 3$

Case II:
$\frac{1}{(k - 1)} = \frac{5}{(3 k + 1)}$
$\Rightarrow 3 k + 1 = 5 (k - 1)$
$\Rightarrow 3 k + 1 = 5 k - 5$
$\Rightarrow 2 k = 6 \Rightarrow k = 3$

Case III:
$\frac{(k - 1)}{(k + 1)} = \frac{5}{(3 k + 1)}$
$\Rightarrow (3 k + 1) (k - 1) = 5 (k + 1)$
$\Rightarrow 3 k^{2} + k - 3 k - 1 = 5 k + 5$
$\Rightarrow 3 k^{2} - 2 k - 5 k - 1 - 5 = 0$
$\Rightarrow 3 k^{2} - 7 k - 6 = 0$
$\Rightarrow 3 k^{2} - 9 k + 2 k - 6 = 0$
$\Rightarrow 3 k (k - 3) + 2 (k - 3) = 0$
$\Rightarrow (k - 3) (3 k + 2) = 0$
$\Rightarrow (k - 3) = 0 or (3 k + 2) = 0$
$\Rightarrow k = 3 or k = \frac{- 2}{3}$

Hence, the given system of equations has an infinite number of solutions when k is equal to 3.

Answer:

The given system of equations can be written as
$(k - 3) x + 3 y - k = 0 . . . . . (i) k x + k y - 12 = 0 . . . . . (i i)$
This system is of the form
$a_{1} x + b_{1} y + c_{1} = 0 a_{2} x + b_{2} y + c_{2} = 0$
where $a_{1} = k - 3, b_{1} = 3, c_{1} = - k and a_{2} = k, b_{2} = k, c_{2} = - 12$
For the given system of linear equations to have an infinite number of solutions
, we must have
$\frac{a_{1}}{a_{2}} = \frac{b_{1}}{b_{2}} = \frac{c_{1}}{c_{2}} \Rightarrow \frac{k - 3}{k} = \frac{3}{k} = \frac{- k}{- 12} \Rightarrow \frac{k - 3}{k} = \frac{3}{k} and \frac{3}{k} = \frac{- k}{- 12} \Rightarrow k - 3 = 3 and k^{2} = 36$
$\Rightarrow k = 6 and k = \pm 6 \Rightarrow k = 6$
Hence, k = 6.

Question 21:

The given system of equations can be written as
$(k - 3) x + 3 y - k = 0 . . . . . (i) k x + k y - 12 = 0 . . . . . (i i)$
This system is of the form
$a_{1} x + b_{1} y + c_{1} = 0 a_{2} x + b_{2} y + c_{2} = 0$
where $a_{1} = k - 3, b_{1} = 3, c_{1} = - k and a_{2} = k, b_{2} = k, c_{2} = - 12$
For the given system of linear equations to have an infinite number of solutions
, we must have
$\frac{a_{1}}{a_{2}} = \frac{b_{1}}{b_{2}} = \frac{c_{1}}{c_{2}} \Rightarrow \frac{k - 3}{k} = \frac{3}{k} = \frac{- k}{- 12} \Rightarrow \frac{k - 3}{k} = \frac{3}{k} and \frac{3}{k} = \frac{- k}{- 12} \Rightarrow k - 3 = 3 and k^{2} = 36$
$\Rightarrow k = 6 and k = \pm 6 \Rightarrow k = 6$
Hence, k = 6.

Answer:

The given system of equations:
(a − 1)x + 3y = 2
⇒ (a − 1)x + 3y − 2 = 0 ...(i)
and 6x + (1 − 2b)y = 6
⇒ 6x + (1 − 2b)y − 6= 0 ...(ii)
These equations are of the following form:
a₁x + b₁y + c₁= 0, a₂x + b₂y + c₂ = 0
where, a₁ = (a − 1), b₁= 3, c₁ = −2 and a₂ = 6, b₂= (1 − 2b), c₂ = −6
For an infinite number of solutions, we must have:
$\frac{a_{1}}{a_{2}} = \frac{b_{1}}{b_{2}} = \frac{c_{1}}{c_{2}}$
$\frac{(a - 1)}{6} = \frac{3}{(1 - 2 b)} = \frac{- 2}{- 6}$
$\Rightarrow \frac{a - 1}{6} = \frac{3}{(1 - 2 b)} = \frac{1}{3}$
$\Rightarrow \frac{a - 1}{6} = \frac{1}{3} and \frac{3}{(1 - 2 b)} = \frac{1}{3}$
⇒ 3a − 3 = 6 and 9 = 1 − 2b
⇒ 3a = 9 and 2b = −8
⇒ a = 3 and b = −4
∴â€‹ a = 3 and b = −4

Question 22:

The given system of equations:
(a − 1)x + 3y = 2
⇒ (a − 1)x + 3y − 2 = 0 ...(i)
and 6x + (1 − 2b)y = 6
⇒ 6x + (1 − 2b)y − 6= 0 ...(ii)
These equations are of the following form:
a₁x + b₁y + c₁= 0, a₂x + b₂y + c₂ = 0
where, a₁ = (a − 1), b₁= 3, c₁ = −2 and a₂ = 6, b₂= (1 − 2b), c₂ = −6
For an infinite number of solutions, we must have:
$\frac{a_{1}}{a_{2}} = \frac{b_{1}}{b_{2}} = \frac{c_{1}}{c_{2}}$
$\frac{(a - 1)}{6} = \frac{3}{(1 - 2 b)} = \frac{- 2}{- 6}$
$\Rightarrow \frac{a - 1}{6} = \frac{3}{(1 - 2 b)} = \frac{1}{3}$
$\Rightarrow \frac{a - 1}{6} = \frac{1}{3} and \frac{3}{(1 - 2 b)} = \frac{1}{3}$
⇒ 3a − 3 = 6 and 9 = 1 − 2b
⇒ 3a = 9 and 2b = −8
⇒ a = 3 and b = −4
∴â€‹ a = 3 and b = −4

Answer:

The given system of equations:
(2a − 1)x + 3y = 5
⇒ (2a − 1)x + 3y − 5 = 0 ....(i)
And, 3x + (b − 1)y = 2
⇒ 3x + (b − 1)y − 2 = 0 ....(ii)
These equations are of the following form:
a₁x + b₁y + c₁= 0, a₂x + b₂y + c₂ = 0
Here, a₁ = (2a − 1), b₁= 3, c₁ = −5 and a₂ = 3, b₂= (b − 1), c₂ = −2
For an infinite number of solutions, we must have:
$\frac{a_{1}}{a_{2}} = \frac{b_{1}}{b_{2}} = \frac{c_{1}}{c_{2}}$
$\frac{(2 a - 1)}{3} = \frac{3}{(b - 1)} = \frac{- 5}{- 2}$
$\Rightarrow \frac{(2 a - 1)}{3} = \frac{3}{(b - 1)} = \frac{5}{2}$
$\Rightarrow \frac{(2 a - 1)}{3} = \frac{5}{2} and \frac{3}{(b - 1)} = \frac{5}{2}$
⇒ 2(2a − 1) = 15 and 6 = 5(b − 1)
⇒ 4a − 2 = 15 and 6 = 5b − 5
⇒ 4a = 17 and 5b = 11
$\Rightarrow a = \frac{17}{4} and b = \frac{11}{5}$
∴â€‹ a = $\frac{17}{4}$ and b = $\frac{11}{5}$

Question 23:

The given system of equations:
(2a − 1)x + 3y = 5
⇒ (2a − 1)x + 3y − 5 = 0 ....(i)
And, 3x + (b − 1)y = 2
⇒ 3x + (b − 1)y − 2 = 0 ....(ii)
These equations are of the following form:
a₁x + b₁y + c₁= 0, a₂x + b₂y + c₂ = 0
Here, a₁ = (2a − 1), b₁= 3, c₁ = −5 and a₂ = 3, b₂= (b − 1), c₂ = −2
For an infinite number of solutions, we must have:
$\frac{a_{1}}{a_{2}} = \frac{b_{1}}{b_{2}} = \frac{c_{1}}{c_{2}}$
$\frac{(2 a - 1)}{3} = \frac{3}{(b - 1)} = \frac{- 5}{- 2}$
$\Rightarrow \frac{(2 a - 1)}{3} = \frac{3}{(b - 1)} = \frac{5}{2}$
$\Rightarrow \frac{(2 a - 1)}{3} = \frac{5}{2} and \frac{3}{(b - 1)} = \frac{5}{2}$
⇒ 2(2a − 1) = 15 and 6 = 5(b − 1)
⇒ 4a − 2 = 15 and 6 = 5b − 5
⇒ 4a = 17 and 5b = 11
$\Rightarrow a = \frac{17}{4} and b = \frac{11}{5}$
∴â€‹ a = $\frac{17}{4}$ and b = $\frac{11}{5}$

Answer:

The given system of equations:
2x − 3y = 7
⇒ 2x − 3y − 7 = 0                                ....(i)
And, (a + b)x − (a + b − 3)y = 4a + b
⇒ (a + b)x − (a + b − 3)y − (4a + b) = 0 ....(ii)
These equations are of the following form:
a₁x + b₁y + c₁= 0, a₂x + b₂y + c₂ = 0
Here, a₁ = 2, b₁= −3, c₁ = −7 and a₂ = (a + b), b₂= −(a + b − 3), c₂ = −(4a + b)
For an infinite number of solutions, we must have:
$\frac{a_{1}}{a_{2}} = \frac{b_{1}}{b_{2}} = \frac{c_{1}}{c_{2}}$
$\frac{2}{a + b} = \frac{- 3}{- (a + b - 3)} = \frac{- 7}{- (4 a + b)}$
$\Rightarrow \frac{2}{a + b} = \frac{3}{(a + b - 3)} = \frac{7}{(4 a + b)}$
$\Rightarrow \frac{2}{a + b} = \frac{7}{(4 a + b)} and \frac{3}{(a + b - 3)} = \frac{7}{(4 a + b)}$
⇒ 2(4a + b) = 7(a + b) and 3(4a + b) = 7(a + b − 3)
⇒ 8a + 2b = 7a + 7b and 12a + 3b = 7a + 7b − 21
⇒ a = 5b                  ....(iii)
And, 5a = 4b − 21     ....(iv)
On substituting a = 5b in (iv), we get:
25b = 4b − 21
⇒ 21b = −21
⇒ b = −1
On substituting b = −1 in (iii), we get:
a = 5(−1) = −5
∴ a = −5 and b = −1

Question 24:

The given system of equations:
2x − 3y = 7
⇒ 2x − 3y − 7 = 0                                ....(i)
And, (a + b)x − (a + b − 3)y = 4a + b
⇒ (a + b)x − (a + b − 3)y − (4a + b) = 0 ....(ii)
These equations are of the following form:
a₁x + b₁y + c₁= 0, a₂x + b₂y + c₂ = 0
Here, a₁ = 2, b₁= −3, c₁ = −7 and a₂ = (a + b), b₂= −(a + b − 3), c₂ = −(4a + b)
For an infinite number of solutions, we must have:
$\frac{a_{1}}{a_{2}} = \frac{b_{1}}{b_{2}} = \frac{c_{1}}{c_{2}}$
$\frac{2}{a + b} = \frac{- 3}{- (a + b - 3)} = \frac{- 7}{- (4 a + b)}$
$\Rightarrow \frac{2}{a + b} = \frac{3}{(a + b - 3)} = \frac{7}{(4 a + b)}$
$\Rightarrow \frac{2}{a + b} = \frac{7}{(4 a + b)} and \frac{3}{(a + b - 3)} = \frac{7}{(4 a + b)}$
⇒ 2(4a + b) = 7(a + b) and 3(4a + b) = 7(a + b − 3)
⇒ 8a + 2b = 7a + 7b and 12a + 3b = 7a + 7b − 21
⇒ a = 5b                  ....(iii)
And, 5a = 4b − 21     ....(iv)
On substituting a = 5b in (iv), we get:
25b = 4b − 21
⇒ 21b = −21
⇒ b = −1
On substituting b = −1 in (iii), we get:
a = 5(−1) = −5
∴ a = −5 and b = −1

Answer:

The given system of equations:
2x + 3y = 7
⇒ 2x + 3y − 7 = 0                                   ....(i)
And, (a + b + 1)x + (a + 2b + 2)y = 4(a + b) + 1
⇒ (a + b + 1)x + (a + 2b + 2)y − [4(a + b) + 1] = 0    ....(ii)
These equations are of the following form:
a₁x + b₁y + c₁= 0, a₂x + b₂y + c₂ = 0
Here, a₁ = 2, b₁= 3, c₁ = −7 and a₂ = (a + b + 1), b₂= (a + 2b + 2), c₂ = −[4(a + b) + 1]
For an infinite number of solutions, we must have:
$\frac{a_{1}}{a_{2}} = \frac{b_{1}}{b_{2}} = \frac{c_{1}}{c_{2}}$
$\frac{2}{(a + b + 1)} = \frac{3}{(a + 2 b + 2)} = \frac{- 7}{- [4 (a + b) + 1]}$
$\Rightarrow \frac{2}{(a + b + 1)} = \frac{3}{(a + 2 b + 2)} = \frac{7}{4 (a + b) + 1}$
$\Rightarrow \frac{2}{(a + b + 1)} = \frac{3}{(a + 2 b + 2)} and \frac{3}{(a + 2 b + 2)} = \frac{7}{4 (a + b) + 1}$

⇒ 2(a + 2b + 2) = 3(a + b+ 1) and 3[4(a + b) + 1] = 7(a + 2b + 2)
⇒ 2a + 4b + 4 = 3a + 3b + 3 and 3(4a + 4b + 1) = 7a + 14b + 14
⇒ a − b − 1 = 0 and 12a + 12b + 3 = 7a + 14b + 14
⇒ a − b = 1 and 5a − 2b = 11
a = (b + 1)     ....(iii)
5a − 2b = 11     ....(iv)
On substituting a = (b + 1) in (iv), we get:
5(b + 1) − 2b = 11
⇒ 5b + 5 − 2b = 11
⇒ 3b = 6
⇒ b = 2
On substituting b = 2 in (iii), we get:
a = 3
∴â€‹ a = 3 and b = 2

Question 25:

The given system of equations:
2x + 3y = 7
⇒ 2x + 3y − 7 = 0                                   ....(i)
And, (a + b + 1)x + (a + 2b + 2)y = 4(a + b) + 1
⇒ (a + b + 1)x + (a + 2b + 2)y − [4(a + b) + 1] = 0    ....(ii)
These equations are of the following form:
a₁x + b₁y + c₁= 0, a₂x + b₂y + c₂ = 0
Here, a₁ = 2, b₁= 3, c₁ = −7 and a₂ = (a + b + 1), b₂= (a + 2b + 2), c₂ = −[4(a + b) + 1]
For an infinite number of solutions, we must have:
$\frac{a_{1}}{a_{2}} = \frac{b_{1}}{b_{2}} = \frac{c_{1}}{c_{2}}$
$\frac{2}{(a + b + 1)} = \frac{3}{(a + 2 b + 2)} = \frac{- 7}{- [4 (a + b) + 1]}$
$\Rightarrow \frac{2}{(a + b + 1)} = \frac{3}{(a + 2 b + 2)} = \frac{7}{4 (a + b) + 1}$
$\Rightarrow \frac{2}{(a + b + 1)} = \frac{3}{(a + 2 b + 2)} and \frac{3}{(a + 2 b + 2)} = \frac{7}{4 (a + b) + 1}$

⇒ 2(a + 2b + 2) = 3(a + b+ 1) and 3[4(a + b) + 1] = 7(a + 2b + 2)
⇒ 2a + 4b + 4 = 3a + 3b + 3 and 3(4a + 4b + 1) = 7a + 14b + 14
⇒ a − b − 1 = 0 and 12a + 12b + 3 = 7a + 14b + 14
⇒ a − b = 1 and 5a − 2b = 11
a = (b + 1)     ....(iii)
5a − 2b = 11     ....(iv)
On substituting a = (b + 1) in (iv), we get:
5(b + 1) − 2b = 11
⇒ 5b + 5 − 2b = 11
⇒ 3b = 6
⇒ b = 2
On substituting b = 2 in (iii), we get:
a = 3
∴â€‹ a = 3 and b = 2

Answer:

The given system of equations can be written as
$2 x + 3 y - 7 = 0 . . . . . (i) (a + b) x + (2 a - b) y - 21 = 0 . . . . . (i i)$
This system is of the form
$a_{1} x + b_{1} y + c_{1} = 0 a_{2} x + b_{2} y + c_{2} = 0$
where $a_{1} = 2, b_{1} = 3, c_{1} = - 7 and a_{2} = a + b, b_{2} = 2 a - b, c_{2} = - 21$
For the given system of linear equations to have an infinite number of solutions, we must have
$\frac{a_{1}}{a_{2}} = \frac{b_{1}}{b_{2}} = \frac{c_{1}}{c_{2}} \Rightarrow \frac{2}{a + b} = \frac{3}{2 a - b} = \frac{- 7}{- 21} \Rightarrow \frac{2}{a + b} = \frac{- 7}{- 21} = \frac{1}{3} and \frac{3}{2 a - b} = \frac{- 7}{- 21} = \frac{1}{3} \Rightarrow a + b = 6 and 2 a - b = 9$
Adding $a + b = 6 and 2 a - b = 9$ , we get
$3 a = 15 \Rightarrow a = \frac{15}{3} = 5$
Now, substituting a = 5 in a + b = 6, we have
$5 + b = 6 \Rightarrow b = 6 - 5 = 1$
Hence, a = 5 and b = 1.

Question 26:

The given system of equations can be written as
$2 x + 3 y - 7 = 0 . . . . . (i) (a + b) x + (2 a - b) y - 21 = 0 . . . . . (i i)$
This system is of the form
$a_{1} x + b_{1} y + c_{1} = 0 a_{2} x + b_{2} y + c_{2} = 0$
where $a_{1} = 2, b_{1} = 3, c_{1} = - 7 and a_{2} = a + b, b_{2} = 2 a - b, c_{2} = - 21$
For the given system of linear equations to have an infinite number of solutions, we must have
$\frac{a_{1}}{a_{2}} = \frac{b_{1}}{b_{2}} = \frac{c_{1}}{c_{2}} \Rightarrow \frac{2}{a + b} = \frac{3}{2 a - b} = \frac{- 7}{- 21} \Rightarrow \frac{2}{a + b} = \frac{- 7}{- 21} = \frac{1}{3} and \frac{3}{2 a - b} = \frac{- 7}{- 21} = \frac{1}{3} \Rightarrow a + b = 6 and 2 a - b = 9$
Adding $a + b = 6 and 2 a - b = 9$ , we get
$3 a = 15 \Rightarrow a = \frac{15}{3} = 5$
Now, substituting a = 5 in a + b = 6, we have
$5 + b = 6 \Rightarrow b = 6 - 5 = 1$
Hence, a = 5 and b = 1.

Answer:

The given system of equations can be written as
$2 x + 3 y - 7 = 0 . . . . . (i) 2 a x + (a + b) y - 28 = 0 . . . . . (i i)$
This system is of the form
$a_{1} x + b_{1} y + c_{1} = 0 a_{2} x + b_{2} y + c_{2} = 0$
where $a_{1} = 2, b_{1} = 3, c_{1} = - 7 and a_{2} = 2 a, b_{2} = a + b, c_{2} = - 28$
For the given system of linear equations to have an infinite number of solutions, we must have
$\frac{a_{1}}{a_{2}} = \frac{b_{1}}{b_{2}} = \frac{c_{1}}{c_{2}} \Rightarrow \frac{2}{2 a} = \frac{3}{a + b} = \frac{- 7}{- 28} \Rightarrow \frac{2}{2 a} = \frac{- 7}{- 28} = \frac{1}{4} and \frac{3}{a + b} = \frac{- 7}{- 28} = \frac{1}{4} \Rightarrow a = 4 and a + b = 12$
Substituting a = 4 in a + b = 12, we get
$4 + b = 12 \Rightarrow b = 12 - 4 = 8$
Hence, a = 4 and b = 8.

Question 27:

The given system of equations can be written as
$2 x + 3 y - 7 = 0 . . . . . (i) 2 a x + (a + b) y - 28 = 0 . . . . . (i i)$
This system is of the form
$a_{1} x + b_{1} y + c_{1} = 0 a_{2} x + b_{2} y + c_{2} = 0$
where $a_{1} = 2, b_{1} = 3, c_{1} = - 7 and a_{2} = 2 a, b_{2} = a + b, c_{2} = - 28$
For the given system of linear equations to have an infinite number of solutions, we must have
$\frac{a_{1}}{a_{2}} = \frac{b_{1}}{b_{2}} = \frac{c_{1}}{c_{2}} \Rightarrow \frac{2}{2 a} = \frac{3}{a + b} = \frac{- 7}{- 28} \Rightarrow \frac{2}{2 a} = \frac{- 7}{- 28} = \frac{1}{4} and \frac{3}{a + b} = \frac{- 7}{- 28} = \frac{1}{4} \Rightarrow a = 4 and a + b = 12$
Substituting a = 4 in a + b = 12, we get
$4 + b = 12 \Rightarrow b = 12 - 4 = 8$
Hence, a = 4 and b = 8.

Answer:

The given system of equations:
8x + 5y = 9
8x + 5y − 9 = 0 ....(i)
kx + 10y = 15
kx + 10y − 15= 0 ....(ii)
These equations are of the following form:
a₁x + b₁y + c₁= 0, a₂x + b₂y + c₂ = 0
Here, a₁ = 8, b₁= 5, c₁ = −9 and a₂ = k, b₂= 10, c₂ = −15
In order that the given system has no solution, we must have:
$\frac{a_{1}}{a_{2}} = \frac{b_{1}}{b_{2}} \neq \frac{c_{1}}{c_{2}}$
$i . e . \frac{8}{k} = \frac{5}{10} \neq \frac{- 9}{- 15} i . e . \frac{8}{k} = \frac{1}{2} \neq \frac{3}{5}$
$\frac{8}{k} = \frac{1}{2}$ and $\frac{8}{k} \neq \frac{3}{5}$
$\Rightarrow k = 16 and k \neq \frac{40}{3}$
Hence, the given system of equations has no solution when k is equal to 16.

Question 28:

The given system of equations:
8x + 5y = 9
8x + 5y − 9 = 0 ....(i)
kx + 10y = 15
kx + 10y − 15= 0 ....(ii)
These equations are of the following form:
a₁x + b₁y + c₁= 0, a₂x + b₂y + c₂ = 0
Here, a₁ = 8, b₁= 5, c₁ = −9 and a₂ = k, b₂= 10, c₂ = −15
In order that the given system has no solution, we must have:
$\frac{a_{1}}{a_{2}} = \frac{b_{1}}{b_{2}} \neq \frac{c_{1}}{c_{2}}$
$i . e . \frac{8}{k} = \frac{5}{10} \neq \frac{- 9}{- 15} i . e . \frac{8}{k} = \frac{1}{2} \neq \frac{3}{5}$
$\frac{8}{k} = \frac{1}{2}$ and $\frac{8}{k} \neq \frac{3}{5}$
$\Rightarrow k = 16 and k \neq \frac{40}{3}$
Hence, the given system of equations has no solution when k is equal to 16.

Answer:

The given system of equations:
kx + 3y = 3
kx + 3y − 3 = 0 ....(i)
12x + ky = 6
12x + ky − 6= 0 ....(ii)
These equations are of the following form:
a₁x + b₁y + c₁= 0, a₂x + b₂y + c₂ = 0
Here, a₁ = k, b₁= 3, c₁ = −3 and a₂ = 12, b₂= k, c₂ = −6
In order that the given system of equations has no solution, we must have:
$\frac{a_{1}}{a_{2}} = \frac{b_{1}}{b_{2}} \neq \frac{c_{1}}{c_{2}}$
i.e. $\frac{k}{12} = \frac{3}{k} \neq \frac{- 3}{- 6}$
$\frac{k}{12} = \frac{3}{k}$ and $\frac{3}{k} \neq \frac{1}{2}$
$\Rightarrow k^{2} = 36 and k \neq 6$
$\Rightarrow k = \pm 6 and k \neq 6$
Hence, the given system of equations has no solution when k is equal to −6.

Question 29:

The given system of equations:
kx + 3y = 3
kx + 3y − 3 = 0 ....(i)
12x + ky = 6
12x + ky − 6= 0 ....(ii)
These equations are of the following form:
a₁x + b₁y + c₁= 0, a₂x + b₂y + c₂ = 0
Here, a₁ = k, b₁= 3, c₁ = −3 and a₂ = 12, b₂= k, c₂ = −6
In order that the given system of equations has no solution, we must have:
$\frac{a_{1}}{a_{2}} = \frac{b_{1}}{b_{2}} \neq \frac{c_{1}}{c_{2}}$
i.e. $\frac{k}{12} = \frac{3}{k} \neq \frac{- 3}{- 6}$
$\frac{k}{12} = \frac{3}{k}$ and $\frac{3}{k} \neq \frac{1}{2}$
$\Rightarrow k^{2} = 36 and k \neq 6$
$\Rightarrow k = \pm 6 and k \neq 6$
Hence, the given system of equations has no solution when k is equal to −6.

Answer:

The given system of equations:
3x − y − 5 = 0 ...(i)
And, 6x − 2y + k = 0 ...(ii)
These equations are of the following form:
a₁x + b₁y + c₁= 0, a₂x + b₂y + c₂ = 0
Here, a₁ = 3, b₁= −1, c₁ = −5 and a₂ = 6, b₂= −2, c₂ = k
In order that the given system of equations has no solution, we must have:
$\frac{a_{1}}{a_{2}} = \frac{b_{1}}{b_{2}} \neq \frac{c_{1}}{c_{2}}$
i.e. $\frac{3}{6} = \frac{- 1}{- 2} \neq \frac{- 5}{k}$
⇒ $\frac{- 1}{- 2} \neq \frac{- 5}{k} \Rightarrow k \neq - 10$
Hence, equations (i) and (ii) will have no solution if $k \neq - 10$ .

Question 30:

The given system of equations:
3x − y − 5 = 0 ...(i)
And, 6x − 2y + k = 0 ...(ii)
These equations are of the following form:
a₁x + b₁y + c₁= 0, a₂x + b₂y + c₂ = 0
Here, a₁ = 3, b₁= −1, c₁ = −5 and a₂ = 6, b₂= −2, c₂ = k
In order that the given system of equations has no solution, we must have:
$\frac{a_{1}}{a_{2}} = \frac{b_{1}}{b_{2}} \neq \frac{c_{1}}{c_{2}}$
i.e. $\frac{3}{6} = \frac{- 1}{- 2} \neq \frac{- 5}{k}$
⇒ $\frac{- 1}{- 2} \neq \frac{- 5}{k} \Rightarrow k \neq - 10$
Hence, equations (i) and (ii) will have no solution if $k \neq - 10$ .

Answer:

The given system of equations can be written as
$k x + 3 y + 3 - k = 0 . . . . . (i) 12 x + k y - k = 0 . . . . . (i i)$
This system is of the form
$a_{1} x + b_{1} y + c_{1} = 0 a_{2} x + b_{2} y + c_{2} = 0$
where $a_{1} = k, b_{1} = 3, c_{1} = 3 - k and a_{2} = 12, b_{2} = k, c_{2} = - k$
For the given system of linear equations to have no solution, we must have
$\frac{a_{1}}{a_{2}} = \frac{b_{1}}{b_{2}} \neq \frac{c_{1}}{c_{2}} \Rightarrow \frac{k}{12} = \frac{3}{k} \neq \frac{3 - k}{- k} \Rightarrow \frac{k}{12} = \frac{3}{k} and \frac{3}{k} \neq \frac{3 - k}{- k} \Rightarrow k^{2} = 36 and - 3 \neq 3 - k$
$\Rightarrow k = \pm 6 and k \neq 6 \Rightarrow k = - 6$
Hence, $k = - 6$ .

Question 31:

The given system of equations can be written as
$k x + 3 y + 3 - k = 0 . . . . . (i) 12 x + k y - k = 0 . . . . . (i i)$
This system is of the form
$a_{1} x + b_{1} y + c_{1} = 0 a_{2} x + b_{2} y + c_{2} = 0$
where $a_{1} = k, b_{1} = 3, c_{1} = 3 - k and a_{2} = 12, b_{2} = k, c_{2} = - k$
For the given system of linear equations to have no solution, we must have
$\frac{a_{1}}{a_{2}} = \frac{b_{1}}{b_{2}} \neq \frac{c_{1}}{c_{2}} \Rightarrow \frac{k}{12} = \frac{3}{k} \neq \frac{3 - k}{- k} \Rightarrow \frac{k}{12} = \frac{3}{k} and \frac{3}{k} \neq \frac{3 - k}{- k} \Rightarrow k^{2} = 36 and - 3 \neq 3 - k$
$\Rightarrow k = \pm 6 and k \neq 6 \Rightarrow k = - 6$
Hence, $k = - 6$ .

Answer:

The given system of equations:
5x − 3y = 0 ....(i)
2x + ky = 0 ....(ii)
These equations are of the following form:
a₁x + b₁y + c₁= 0, a₂x + b₂y + c₂ = 0
Here, a₁ = 5, b₁= −3, c₁ = 0 and a₂ = 2, b₂= k, c₂ = 0
For a non-zero solution, we must have:
$\frac{a_{1}}{a_{2}} = \frac{b_{1}}{b_{2}}$
$\Rightarrow \frac{5}{2} = \frac{- 3}{k}$
$\Rightarrow 5 k = - 6 \Rightarrow k = \frac{- 6}{5}$
Hence, the required value of k is $\frac{- 6}{5}$ .

Question 1:

The given system of equations:
5x − 3y = 0 ....(i)
2x + ky = 0 ....(ii)
These equations are of the following form:
a₁x + b₁y + c₁= 0, a₂x + b₂y + c₂ = 0
Here, a₁ = 5, b₁= −3, c₁ = 0 and a₂ = 2, b₂= k, c₂ = 0
For a non-zero solution, we must have:
$\frac{a_{1}}{a_{2}} = \frac{b_{1}}{b_{2}}$
$\Rightarrow \frac{5}{2} = \frac{- 3}{k}$
$\Rightarrow 5 k = - 6 \Rightarrow k = \frac{- 6}{5}$
Hence, the required value of k is $\frac{- 6}{5}$ .

Answer:

Let the cost of a chair be â‚¹ x and that of a table be â‚¹ y. Then
$5 x + 4 y = 5600 . . . . . (i) 4 x + 3 y = 4340 . . . . . (i i)$
Multiplying (i) by 3 and (ii) by 4, we get
$15 x - 16 x = 16800 - 17360 \Rightarrow - x = - 560 \Rightarrow x = 560$
Substituting x = 560 in (i), we have
$5 \times 560 + 4 y = 5600 \Rightarrow 4 y = 5600 - 2800 \Rightarrow y = \frac{2800}{4} = 700$
Hence, the cost of a chair and that of a table are respectively â‚¹ 560 and â‚¹ 700.

Question 2:

Let the cost of a chair be â‚¹ x and that of a table be â‚¹ y. Then
$5 x + 4 y = 5600 . . . . . (i) 4 x + 3 y = 4340 . . . . . (i i)$
Multiplying (i) by 3 and (ii) by 4, we get
$15 x - 16 x = 16800 - 17360 \Rightarrow - x = - 560 \Rightarrow x = 560$
Substituting x = 560 in (i), we have
$5 \times 560 + 4 y = 5600 \Rightarrow 4 y = 5600 - 2800 \Rightarrow y = \frac{2800}{4} = 700$
Hence, the cost of a chair and that of a table are respectively â‚¹ 560 and â‚¹ 700.

Answer:

Let the cost of a spoon be â‚¹x and that of a fork be â‚¹y. Then
$23 x + 17 y = 1770 . . . . . (i) 17 x + 23 y = 1830 . . . . . (i i)$
Adding (i) and (ii), we get
$40 x + 40 y = 3600 \Rightarrow x + y = 90 . . . . . (i i i)$
Now, subtracting (ii) from (i), we get
$6 x - 6 y = - 60 \Rightarrow x - y = - 10 . . . . . (i v)$
Adding (iii) and (iv), we get
$2 x = 80 \Rightarrow x = 40$
Substituting x = 40 in (iii), we get
$40 + y = 90 \Rightarrow y = 50$
Hence, the cost of a spoon that of a fork are â‚¹40 and â‚¹50 respectively.

Question 3:

Let the cost of a spoon be â‚¹x and that of a fork be â‚¹y. Then
$23 x + 17 y = 1770 . . . . . (i) 17 x + 23 y = 1830 . . . . . (i i)$
Adding (i) and (ii), we get
$40 x + 40 y = 3600 \Rightarrow x + y = 90 . . . . . (i i i)$
Now, subtracting (ii) from (i), we get
$6 x - 6 y = - 60 \Rightarrow x - y = - 10 . . . . . (i v)$
Adding (iii) and (iv), we get
$2 x = 80 \Rightarrow x = 40$
Substituting x = 40 in (iii), we get
$40 + y = 90 \Rightarrow y = 50$
Hence, the cost of a spoon that of a fork are â‚¹40 and â‚¹50 respectively.

Answer:

Let the x and y be the number of 50-paisa and 25-paisa conis respectively. Then
$x + y = 50 . . . . . (i) 0.5 x + 0.25 y = 19.50 . . . . . (i i)$
Multiplying (ii) by 2 and subtracting it from (i), we get
$0.5 y = 50 - 39 \Rightarrow y = \frac{11}{0.5} = 22$
Subtracting y = 22 in (i), we get
$x + 22 = 50 \Rightarrow x = 50 - 22 = 28$
Hence, the number of 25-paisa and 50-paisa conis are 22 and 28 respectively.

Question 4:

Let the x and y be the number of 50-paisa and 25-paisa conis respectively. Then
$x + y = 50 . . . . . (i) 0.5 x + 0.25 y = 19.50 . . . . . (i i)$
Multiplying (ii) by 2 and subtracting it from (i), we get
$0.5 y = 50 - 39 \Rightarrow y = \frac{11}{0.5} = 22$
Subtracting y = 22 in (i), we get
$x + 22 = 50 \Rightarrow x = 50 - 22 = 28$
Hence, the number of 25-paisa and 50-paisa conis are 22 and 28 respectively.

Answer:

Let the larger number be x and the smaller number be y.
Then, we have:
x + y = 137 ...(i)
x − y = 43 ...(ii)
On adding (i) and (ii), we get:
2x = 180 ⇒ x = 90
On substituting x = 90 in (i), we get:
90 + y = 137
⇒ y = (137 − 90) = 47
Hence, the required numbers are 90 and 47.

Question 5:

Let the larger number be x and the smaller number be y.
Then, we have:
x + y = 137 ...(i)
x − y = 43 ...(ii)
On adding (i) and (ii), we get:
2x = 180 ⇒ x = 90
On substituting x = 90 in (i), we get:
90 + y = 137
⇒ y = (137 − 90) = 47
Hence, the required numbers are 90 and 47.

Answer:

Let the first number be x and the second number be y.
Then, we have:
2x + 3y = 92                       ....(i)
4x − 7y = 2                         ....(ii)
On multiplying (i) by 7 and (ii) by 3, we get:
14x + 21y = 644                 ....(iii)
12x − 21y = 6                     ....(iv)
On adding (iii) and (iv), we get:
26x = 650
⇒ x = 25
On substituting x = 25 in (i), we get:
2 × 25 + 3y = 92
⇒ 50 + 3y = 92
⇒ 3y = (92 − 50) = 42
⇒ y = 14
Hence, the first number is 25 and the second number is 14.

Question 6:

Let the first number be x and the second number be y.
Then, we have:
2x + 3y = 92                       ....(i)
4x − 7y = 2                         ....(ii)
On multiplying (i) by 7 and (ii) by 3, we get:
14x + 21y = 644                 ....(iii)
12x − 21y = 6                     ....(iv)
On adding (iii) and (iv), we get:
26x = 650
⇒ x = 25
On substituting x = 25 in (i), we get:
2 × 25 + 3y = 92
⇒ 50 + 3y = 92
⇒ 3y = (92 − 50) = 42
⇒ y = 14
Hence, the first number is 25 and the second number is 14.

Answer:

Let the first number be x and the second number be y.
Then, we have:
3x + y = 142 ....(i)
4x − y = 138 ....(ii)
On adding (i) and (ii), we get:
7x = 280
⇒ x = 40
On substituting x = 40 in (i), we get:
3 × 40 + y = 142
⇒ y = (142 − 120) = 22
⇒ y = 22
Hence, the first number is 40 and the second number is 22.

Question 7:

Let the first number be x and the second number be y.
Then, we have:
3x + y = 142 ....(i)
4x − y = 138 ....(ii)
On adding (i) and (ii), we get:
7x = 280
⇒ x = 40
On substituting x = 40 in (i), we get:
3 × 40 + y = 142
⇒ y = (142 − 120) = 22
⇒ y = 22
Hence, the first number is 40 and the second number is 22.

Answer:

Let the greater number be x and the smaller be y.
Then, we have:
2x − 45 = y or 2x − y = 45             .... (i)
2y − 21 = x or −x + 2y = 21           ....(ii)
On multiplying (i) by 2, we get:
4x − 2y = 90                                      ....(iii)
On adding (ii) and (iii), we get:
3x = (90 + 21) = 111
⇒ x = 37
On substituting x = 37 in (i), we get:
2 × 37 − y = 45
⇒ 74 − y = 45
⇒ y = (74 − 45) = 29
Hence, the greater number is 37 and the smaller number is 29.

Question 8:

Let the greater number be x and the smaller be y.
Then, we have:
2x − 45 = y or 2x − y = 45             .... (i)
2y − 21 = x or −x + 2y = 21           ....(ii)
On multiplying (i) by 2, we get:
4x − 2y = 90                                      ....(iii)
On adding (ii) and (iii), we get:
3x = (90 + 21) = 111
⇒ x = 37
On substituting x = 37 in (i), we get:
2 × 37 − y = 45
⇒ 74 − y = 45
⇒ y = (74 − 45) = 29
Hence, the greater number is 37 and the smaller number is 29.

Answer:

We know:
Dividend = Divisor × Quotient + Remainder

Let the larger number be x and the smaller be y.
Then, we have:
3x = y × 4 + 8 or 3x − 4y = 8 ....(i)
5y = x × 3 + 5 or −3x + 5y = 5 ....(ii)
On adding (i) and (ii), we get:
y = (8 + 5) = 13
On substituting y = 13 in (i) we get:
3x − 4 × 13 = 8
⇒ 3x = (8 + 52) = 60
⇒ x = 20
Hence, the larger number is 20 and the smaller number is 13.

Question 9:

We know:
Dividend = Divisor × Quotient + Remainder

Let the larger number be x and the smaller be y.
Then, we have:
3x = y × 4 + 8 or 3x − 4y = 8 ....(i)
5y = x × 3 + 5 or −3x + 5y = 5 ....(ii)
On adding (i) and (ii), we get:
y = (8 + 5) = 13
On substituting y = 13 in (i) we get:
3x − 4 × 13 = 8
⇒ 3x = (8 + 52) = 60
⇒ x = 20
Hence, the larger number is 20 and the smaller number is 13.

Answer:

Let the required numbers be x and y.
Now, we have:
$\frac{x + 2}{y + 2} = \frac{1}{2}$
By cross multiplication, we get:
2x + 4 = y + 2
⇒ 2x − y = −2                  ....(i)
Again, we have:
$\frac{x - 4}{y - 4} = \frac{5}{11}$
By cross multiplication, we get:
11x − 44 = 5y − 20
⇒ 11x − 5y = 24                ....(ii)
On multiplying (i) by 5, we get:
10x − 5y = −10                 ....(iii)
On subtracting (iii) from (ii), we get:
x = (24 + 10) = 34
On substituting x = 34 in (i), we get:
2 × 34 − y = −2
⇒ 68 − y = −2
⇒ y = (68 + 2) = 70
Hence, the required numbers are 34 and 70.

Question 10:

Let the required numbers be x and y.
Now, we have:
$\frac{x + 2}{y + 2} = \frac{1}{2}$
By cross multiplication, we get:
2x + 4 = y + 2
⇒ 2x − y = −2                  ....(i)
Again, we have:
$\frac{x - 4}{y - 4} = \frac{5}{11}$
By cross multiplication, we get:
11x − 44 = 5y − 20
⇒ 11x − 5y = 24                ....(ii)
On multiplying (i) by 5, we get:
10x − 5y = −10                 ....(iii)
On subtracting (iii) from (ii), we get:
x = (24 + 10) = 34
On substituting x = 34 in (i), we get:
2 × 34 − y = −2
⇒ 68 − y = −2
⇒ y = (68 + 2) = 70
Hence, the required numbers are 34 and 70.

Answer:

Let the larger number be x and the smaller number be y.
Then, we have:
x − y = 14 or x = 14 + y ....(i)
x² − y² = 448 ....(ii)
On substituting x = 14 + y in (ii), we get:
(14 + y)² − y² = 448
⇒ 196 + y² + 28y − y² = 448
⇒ 196 + 28y = 448
⇒ 28y = (448 − 196) = 252
⇒ $y = \frac{252}{28} = 9$
On substituting y = 9 in (i), we get:
x = 14 + 9 = 23
Hence, the required numbers are 23 and 9.

Question 11:

Let the larger number be x and the smaller number be y.
Then, we have:
x − y = 14 or x = 14 + y ....(i)
x² − y² = 448 ....(ii)
On substituting x = 14 + y in (ii), we get:
(14 + y)² − y² = 448
⇒ 196 + y² + 28y − y² = 448
⇒ 196 + 28y = 448
⇒ 28y = (448 − 196) = 252
⇒ $y = \frac{252}{28} = 9$
On substituting y = 9 in (i), we get:
x = 14 + 9 = 23
Hence, the required numbers are 23 and 9.

Answer:

Let the tens and the units digits of the required number be x and yâ€‹, respectively.
Required number = (10x + y)
x + y = 12 ....(i)
Number obtained on reversing its digits = (10y + x)
∴ (10y + x) − (10x + y) = 18
⇒ 10y + x − 10x − y = 18
⇒ 9y − 9x = 18
⇒ y − x = 2 ....(ii)
On adding (i) and (ii), we get:
2y = 14
⇒ y = 7
On substituting y = 7 in (i), we get:
x + 7 = 12
⇒ x = (12 − 7) = 5
Number = (10x + y) = 10 × 5 + 7 = 50 + 7 = 57
Hence, the required number is 57.

Question 12:

Let the tens and the units digits of the required number be x and yâ€‹, respectively.
Required number = (10x + y)
x + y = 12 ....(i)
Number obtained on reversing its digits = (10y + x)
∴ (10y + x) − (10x + y) = 18
⇒ 10y + x − 10x − y = 18
⇒ 9y − 9x = 18
⇒ y − x = 2 ....(ii)
On adding (i) and (ii), we get:
2y = 14
⇒ y = 7
On substituting y = 7 in (i), we get:
x + 7 = 12
⇒ x = (12 − 7) = 5
Number = (10x + y) = 10 × 5 + 7 = 50 + 7 = 57
Hence, the required number is 57.

Answer:

Let the tens and the units digits of the required number be x and yâ€‹, respectively.
Required number = (10x + y)
10x + y = 7(x + y)
10x + y = 7x + 7y or 3x − 6y = 0       ....(i)
Number obtained on reversing its digits = (10y + x)
(10x + y) − 27 = 10y + x
⇒ 10x − x + y − 10y = 27
⇒ 9x − 9y = 27
⇒ 9(x − y) = 27
⇒ x − y = 3                        ....(ii)
On multiplying (ii) by 6, we get:
6x − 6y = 18                      ....(iii)
On subtracting (i) from (iii), we get:
3x = 18
⇒ x = 6
On substituting x = 6 in (i), we get:
3 × 6 − 6y = 0
⇒ 18 − 6y = 0
⇒ 6y = 18
⇒ y = 3
Number = (10x + y) = 10 × 6 + 3 = 60 + 3 = 63
Hence, the required number is 63.

Question 13:

Let the tens and the units digits of the required number be x and yâ€‹, respectively.
Required number = (10x + y)
10x + y = 7(x + y)
10x + y = 7x + 7y or 3x − 6y = 0       ....(i)
Number obtained on reversing its digits = (10y + x)
(10x + y) − 27 = 10y + x
⇒ 10x − x + y − 10y = 27
⇒ 9x − 9y = 27
⇒ 9(x − y) = 27
⇒ x − y = 3                        ....(ii)
On multiplying (ii) by 6, we get:
6x − 6y = 18                      ....(iii)
On subtracting (i) from (iii), we get:
3x = 18
⇒ x = 6
On substituting x = 6 in (i), we get:
3 × 6 − 6y = 0
⇒ 18 − 6y = 0
⇒ 6y = 18
⇒ y = 3
Number = (10x + y) = 10 × 6 + 3 = 60 + 3 = 63
Hence, the required number is 63.

Answer:

Let the tens and the units digits of the required number be x and yâ€‹, respectively.
Required number = (10x + y)
x + y = 15 ....(i)
Number obtained on reversing its digits = (10y + x)
∴ (10y + x) − (10x + y) = 9
⇒ 10y + x − 10x − y = 9
⇒ 9y − 9x = 9
⇒ y − x = 1 ....(ii)
On adding (i) and (ii), we get:
2y = 16
⇒ y = 8
On substituting y = 8 in (i), we get:
x + 8 = 15
⇒ x = (15 − 8) = 7
Number = (10x + y) = 10 × 7 + 8 = 70 + 8 = 78
Hence, the required number is 78.

Question 14:

Let the tens and the units digits of the required number be x and yâ€‹, respectively.
Required number = (10x + y)
x + y = 15 ....(i)
Number obtained on reversing its digits = (10y + x)
∴ (10y + x) − (10x + y) = 9
⇒ 10y + x − 10x − y = 9
⇒ 9y − 9x = 9
⇒ y − x = 1 ....(ii)
On adding (i) and (ii), we get:
2y = 16
⇒ y = 8
On substituting y = 8 in (i), we get:
x + 8 = 15
⇒ x = (15 − 8) = 7
Number = (10x + y) = 10 × 7 + 8 = 70 + 8 = 78
Hence, the required number is 78.

Answer:

Let the tens and the units digits of the required number be x and yâ€‹, respectively.
Required number = (10x + y)
∴ 10x + y = 4(x + y) + 3
⇒10x + y = 4x + 4y + 3
⇒ 6x − 3y = 3
⇒ 2x − y = 1 ....(i)

Again, we have:
10x + y + 18 = 10y + x
⇒ 9x − 9y = −18
⇒ x − y = −2 ....(ii)
On subtracting (ii) from (i), we get:
x = 3
On substituting x = 3 in (i), we get:
2 × 3 − y = 1
⇒ y = 6 − 1 = 5
Required number = (10x + y) = 10 × 3 + 5 = 30 + 5 = 35
Hence, the required number is 35.

Question 15:

Let the tens and the units digits of the required number be x and yâ€‹, respectively.
Required number = (10x + y)
∴ 10x + y = 4(x + y) + 3
⇒10x + y = 4x + 4y + 3
⇒ 6x − 3y = 3
⇒ 2x − y = 1 ....(i)

Again, we have:
10x + y + 18 = 10y + x
⇒ 9x − 9y = −18
⇒ x − y = −2 ....(ii)
On subtracting (ii) from (i), we get:
x = 3
On substituting x = 3 in (i), we get:
2 × 3 − y = 1
⇒ y = 6 − 1 = 5
Required number = (10x + y) = 10 × 3 + 5 = 30 + 5 = 35
Hence, the required number is 35.

Answer:

We know:
Dividend = (Divisor × Quotient) + Remainder

Let the tens and the units digits of the required number be x and yâ€‹, respectively.
Required number = (10x + y)
∴ 10x + y = (x + y) × 6 + 0
⇒ 10x − 6x + y − 6y = 0
⇒ 4x − 5y = 0                     ....(i)
Number obtained on reversing its digits = (10y + x)
∴ 10x + y − 9 = 10y + x
⇒ 9x − 9y = 9
⇒ 9(x − y) = 9
⇒ x − y = 1                        ....(ii)
On multiplying (ii) by 5, we get:
5x − 5y = 5                        ....(iii)
On subtracting (i) from (iii), we get:
x = 5
On substituting x = 5 in (i), we get:
4 × 5 − 5y = 0
⇒ 20 − 5y = 0
⇒ y = 4
∴ Number = (10x + y) = 10 × 5 + 4 = 50 + 4 = 54
Hence, the required number is 54.

Question 16:

We know:
Dividend = (Divisor × Quotient) + Remainder

Let the tens and the units digits of the required number be x and yâ€‹, respectively.
Required number = (10x + y)
∴ 10x + y = (x + y) × 6 + 0
⇒ 10x − 6x + y − 6y = 0
⇒ 4x − 5y = 0                     ....(i)
Number obtained on reversing its digits = (10y + x)
∴ 10x + y − 9 = 10y + x
⇒ 9x − 9y = 9
⇒ 9(x − y) = 9
⇒ x − y = 1                        ....(ii)
On multiplying (ii) by 5, we get:
5x − 5y = 5                        ....(iii)
On subtracting (i) from (iii), we get:
x = 5
On substituting x = 5 in (i), we get:
4 × 5 − 5y = 0
⇒ 20 − 5y = 0
⇒ y = 4
∴ Number = (10x + y) = 10 × 5 + 4 = 50 + 4 = 54
Hence, the required number is 54.

Answer:

Let the tens and the units digits of the required number be x and yâ€‹, respectively.
Then, we have:
xy = 35                      ....(i)
Required number = (10x + y)
Number obtained on reversing its digits = (10y + x)
∴ (10x + y) + 18 = 10y + x
⇒ 9x − 9y = −18
⇒ 9(y − x) = 18
⇒ y − x = 2                 ....(ii)

We know:
(y + x)² − (y − x)² = 4xy
⇒ $(y + x) = \pm \sqrt{{(y - x)}^{2} + 4 x y}$
⇒ $(y + x) = \pm \sqrt{4 + 4 \times 35} = \pm \sqrt{144} = \pm 12$
∴ y + x = 12              .....(iii) (âˆµ x and y cannot be negative)
On adding (ii) and (iii), we get:
2y = 2 + 12 = 14
⇒ y = 7
On substituting y = 7 in (ii), we get:
7 − x = 2
⇒ x = (7 − 2) = 5
∴ Number = (10x + y) = 10 × 5 + 7 = 50 + 7 = 57
Hence, the required number is 57.

Question 17:

Let the tens and the units digits of the required number be x and yâ€‹, respectively.
Then, we have:
xy = 35                      ....(i)
Required number = (10x + y)
Number obtained on reversing its digits = (10y + x)
∴ (10x + y) + 18 = 10y + x
⇒ 9x − 9y = −18
⇒ 9(y − x) = 18
⇒ y − x = 2                 ....(ii)

We know:
(y + x)² − (y − x)² = 4xy
⇒ $(y + x) = \pm \sqrt{{(y - x)}^{2} + 4 x y}$
⇒ $(y + x) = \pm \sqrt{4 + 4 \times 35} = \pm \sqrt{144} = \pm 12$
∴ y + x = 12              .....(iii) (âˆµ x and y cannot be negative)
On adding (ii) and (iii), we get:
2y = 2 + 12 = 14
⇒ y = 7
On substituting y = 7 in (ii), we get:
7 − x = 2
⇒ x = (7 − 2) = 5
∴ Number = (10x + y) = 10 × 5 + 7 = 50 + 7 = 57
Hence, the required number is 57.

Answer:

Let the tens and the units digits of the required number be x and yâ€‹, respectively.
Then, we have:
xy = 18                       ....(i)
Required number = (10x + y)
Number obtained on reversing its digits = (10y + x)
∴ (10x + y) − 63 = 10y + x
⇒ 9x − 9y = 63
⇒ 9(x − y) = 63
⇒ x − y = 7        ....(ii)

We know:
(x + y)² − (x − y)² = 4xy
⇒ $(x + y) = \pm \sqrt{{(x - y)}^{2} + 4 x y}$
$\Rightarrow (x + y) = \pm \sqrt{49 + 4 \times 18} = \pm \sqrt{49 + 72} = \pm \sqrt{121} = \pm 11$
∴ x + y = 11              ....(iii) (âˆµ x and y cannot be negative)
On adding (ii) and (iii), we get:
2x = 7 + 11 = 18
⇒ x = 9
On substituting x = 9 in (ii), we get:
9 − y = 7
⇒ y = (9 − 7) = 2
∴ Number = (10x + y) = 10 × 9 + 2 = 90 + 2 = 92
Hence, the required number is 92.

Question 18:

Let the tens and the units digits of the required number be x and yâ€‹, respectively.
Then, we have:
xy = 18                       ....(i)
Required number = (10x + y)
Number obtained on reversing its digits = (10y + x)
∴ (10x + y) − 63 = 10y + x
⇒ 9x − 9y = 63
⇒ 9(x − y) = 63
⇒ x − y = 7        ....(ii)

We know:
(x + y)² − (x − y)² = 4xy
⇒ $(x + y) = \pm \sqrt{{(x - y)}^{2} + 4 x y}$
$\Rightarrow (x + y) = \pm \sqrt{49 + 4 \times 18} = \pm \sqrt{49 + 72} = \pm \sqrt{121} = \pm 11$
∴ x + y = 11              ....(iii) (âˆµ x and y cannot be negative)
On adding (ii) and (iii), we get:
2x = 7 + 11 = 18
⇒ x = 9
On substituting x = 9 in (ii), we get:
9 − y = 7
⇒ y = (9 − 7) = 2
∴ Number = (10x + y) = 10 × 9 + 2 = 90 + 2 = 92
Hence, the required number is 92.

Answer:

Let x be the ones digit and y be the tens digit. Then
Two digit number before reversing = 10y + x
Two digit number after reversing = 10x + y
As per the question
$(10 y + x) + (10 x + y) = 121 \Rightarrow 11 x + 11 y = 121 \Rightarrow x + y = 11 . . . . . (i)$
Since the digits differ by 3, so
$x - y = 3 . . . . . . (i i)$
Adding (i) and (ii), we get
$2 x = 14 \Rightarrow x = 7$
Putting x = 7 in (i), we get
$7 + y = 11 \Rightarrow y = 4$
Changing the role of x and y, x = 4 and y = 7
Hence, the two-digit number is 74 or 47.

Question 19:

Let x be the ones digit and y be the tens digit. Then
Two digit number before reversing = 10y + x
Two digit number after reversing = 10x + y
As per the question
$(10 y + x) + (10 x + y) = 121 \Rightarrow 11 x + 11 y = 121 \Rightarrow x + y = 11 . . . . . (i)$
Since the digits differ by 3, so
$x - y = 3 . . . . . . (i i)$
Adding (i) and (ii), we get
$2 x = 14 \Rightarrow x = 7$
Putting x = 7 in (i), we get
$7 + y = 11 \Rightarrow y = 4$
Changing the role of x and y, x = 4 and y = 7
Hence, the two-digit number is 74 or 47.

Answer:

Let the required fraction be $\frac{x}{y}$ .
Then, we have:
x + y = 8                        ....(i)
And, $\frac{x + 3}{y + 3} = \frac{3}{4}$
⇒ 4(x + 3) = 3(y + 3)
⇒ 4x + 12 = 3y + 9
⇒ 4x − 3y = −3              ....(ii)
On multiplying (i) by 3, we get:
3x + 3y = 24                  ....(iii)
On adding (ii) and (iii), we get:
7x = 21
⇒ x = 3
On substituting x = 3 in (i), we get:
3 + y = 8
⇒ y = (8 − 3) = 5
∴â€‹ â€‹x = 3 and y = 5
Hence, the required fraction is $\frac{3}{5}$ .

Question 20:

Let the required fraction be $\frac{x}{y}$ .
Then, we have:
x + y = 8                        ....(i)
And, $\frac{x + 3}{y + 3} = \frac{3}{4}$
⇒ 4(x + 3) = 3(y + 3)
⇒ 4x + 12 = 3y + 9
⇒ 4x − 3y = −3              ....(ii)
On multiplying (i) by 3, we get:
3x + 3y = 24                  ....(iii)
On adding (ii) and (iii), we get:
7x = 21
⇒ x = 3
On substituting x = 3 in (i), we get:
3 + y = 8
⇒ y = (8 − 3) = 5
∴â€‹ â€‹x = 3 and y = 5
Hence, the required fraction is $\frac{3}{5}$ .

Answer:

Let the required fraction be $\frac{x}{y}$ .
Then, we have:
$\frac{x + 2}{y} = \frac{1}{2}$
⇒ 2(x + 2) = y
⇒ 2x + 4 = y
⇒ 2x − y = −4 .....(i)

Again, $\frac{x}{y - 1} = \frac{1}{3}$
⇒ 3x = 1(y − 1)
⇒ 3x − y = −1 .....(ii)
On subtracting (i) from (ii), we get:
x = (−1 + 4) = 3
On substituting x = 3 in (i), we get:
2 × 3 − y = −4
⇒ 6 − y = −4
⇒ y = (6 + 4) = 10
∴ x = 3 and y = 10
Hence, the required fraction is $\frac{3}{10}$ .

Question 21:

Let the required fraction be $\frac{x}{y}$ .
Then, we have:
$\frac{x + 2}{y} = \frac{1}{2}$
⇒ 2(x + 2) = y
⇒ 2x + 4 = y
⇒ 2x − y = −4 .....(i)

Again, $\frac{x}{y - 1} = \frac{1}{3}$
⇒ 3x = 1(y − 1)
⇒ 3x − y = −1 .....(ii)
On subtracting (i) from (ii), we get:
x = (−1 + 4) = 3
On substituting x = 3 in (i), we get:
2 × 3 − y = −4
⇒ 6 − y = −4
⇒ y = (6 + 4) = 10
∴ x = 3 and y = 10
Hence, the required fraction is $\frac{3}{10}$ .

Answer:

Let the required fraction be $\frac{x}{y}$ .
Then, we have:
y = x + 11
⇒ y − x = 11                      ....(i)
Again, $\frac{x + 8}{y + 8} = \frac{3}{4}$
⇒ 4(x + 8) =3(y + 8)
⇒ 4x + 32 = 3y + 24
⇒ 4x − 3y = −8                    ....(ii)
On multiplying (i) by 4, we get:
4y − 4x = 44                        ....(iii)
On adding (ii) and (iii), we get:
y = (−8 + 44) = 36
On substituting y = 36 in (i), we get:
36 − x = 11
⇒ x = (36 − 11) = 25
∴â€‹ â€‹x = 25 and y = 36
Hence, the required fraction is $\frac{25}{36}$ .

Question 22:

Let the required fraction be $\frac{x}{y}$ .
Then, we have:
y = x + 11
⇒ y − x = 11                      ....(i)
Again, $\frac{x + 8}{y + 8} = \frac{3}{4}$
⇒ 4(x + 8) =3(y + 8)
⇒ 4x + 32 = 3y + 24
⇒ 4x − 3y = −8                    ....(ii)
On multiplying (i) by 4, we get:
4y − 4x = 44                        ....(iii)
On adding (ii) and (iii), we get:
y = (−8 + 44) = 36
On substituting y = 36 in (i), we get:
36 − x = 11
⇒ x = (36 − 11) = 25
∴â€‹ â€‹x = 25 and y = 36
Hence, the required fraction is $\frac{25}{36}$ .

Answer:

Let the required fraction be $\frac{x}{y}$ .
Then, we have:
$\frac{x - 1}{y + 2} = \frac{1}{2}$
⇒ 2(x − 1) = 1(y + 2)
⇒ 2x − 2 = y + 2
⇒ 2x − y = 4 ....(i)

Again, $\frac{x - 7}{y - 2} = \frac{1}{3}$
⇒ 3(x − 7) = 1(y − 2)
⇒ 3x − 21 = y − 2
⇒ 3x − y = 19 ....(ii)
On subtracting (i) from (ii), we get:
x = (19 − 4) = 15
On substituting x = 15 in (i), we get:
2 × 15 − y = 4
⇒ 30 − y = 4
⇒ y = 26
∴â€‹ â€‹x = 15 and y = 26
Hence, the given fraction is $\frac{15}{26}$ .

Question 23:

Let the required fraction be $\frac{x}{y}$ .
Then, we have:
$\frac{x - 1}{y + 2} = \frac{1}{2}$
⇒ 2(x − 1) = 1(y + 2)
⇒ 2x − 2 = y + 2
⇒ 2x − y = 4 ....(i)

Again, $\frac{x - 7}{y - 2} = \frac{1}{3}$
⇒ 3(x − 7) = 1(y − 2)
⇒ 3x − 21 = y − 2
⇒ 3x − y = 19 ....(ii)
On subtracting (i) from (ii), we get:
x = (19 − 4) = 15
On substituting x = 15 in (i), we get:
2 × 15 − y = 4
⇒ 30 − y = 4
⇒ y = 26
∴â€‹ â€‹x = 15 and y = 26
Hence, the given fraction is $\frac{15}{26}$ .

Answer:

Let the fraction be $\frac{x}{y}$ .
As per the question
$x + y = 4 + 2 x \Rightarrow y - x = 4 . . . . . (i)$
After changing the numerator and denominator
New numerator = x + 3
New denominator = y + 3
Therefore
$\frac{x + 3}{y + 3} = \frac{2}{3} \Rightarrow 3 (x + 3) = 2 (y + 3) \Rightarrow 3 x + 9 = 2 y + 6 \Rightarrow 2 y - 3 x = 3 . . . . . (i i)$
Multiplying (i) by 3 and subtracting (ii), we get
$3 y - 2 y = 12 - 3 \Rightarrow y = 9$
Now, putting y = 9 in (i), we get
$9 - x = 4 \Rightarrow x = 9 - 4 = 5$
Hence, the fraction is $\frac{5}{9}$ .

Question 24:

Let the fraction be $\frac{x}{y}$ .
As per the question
$x + y = 4 + 2 x \Rightarrow y - x = 4 . . . . . (i)$
After changing the numerator and denominator
New numerator = x + 3
New denominator = y + 3
Therefore
$\frac{x + 3}{y + 3} = \frac{2}{3} \Rightarrow 3 (x + 3) = 2 (y + 3) \Rightarrow 3 x + 9 = 2 y + 6 \Rightarrow 2 y - 3 x = 3 . . . . . (i i)$
Multiplying (i) by 3 and subtracting (ii), we get
$3 y - 2 y = 12 - 3 \Rightarrow y = 9$
Now, putting y = 9 in (i), we get
$9 - x = 4 \Rightarrow x = 9 - 4 = 5$
Hence, the fraction is $\frac{5}{9}$ .

Answer:

Let the larger number be x and the smaller number be y.
Then, we have:
(x + y) = 16                      ....(i)
And, $\frac{1}{x} + \frac{1}{y} = \frac{1}{3}$               ....(ii)
⇒ $\frac{x + y}{x y} = \frac{1}{3}$
⇒ 3(x + y) = xy
⇒ 3 × 16 = xy      [Since from (i), we have: x + y = 16]
∴ xy = 48                     ....(iii)
We know:
(x − y)² = (x + y)² − 4xy
(x − y)² = (16)² − 4 × 48 = 256 − 192 = 64
∴ (x − y) = $\pm \sqrt{64} = \pm 8$
Since x is larger and y is smaller, we have:
x − y = 8           .....(iv)
On adding (i) and (iv), we get:
2x = 24
⇒ x = 12
On substituting x = 12 in (i), we get:
12 + y = 16 ⇒ y = (16 − 12) = 4
Hence, the required numbers are 12 and 4.

Question 25:

Let the larger number be x and the smaller number be y.
Then, we have:
(x + y) = 16                      ....(i)
And, $\frac{1}{x} + \frac{1}{y} = \frac{1}{3}$               ....(ii)
⇒ $\frac{x + y}{x y} = \frac{1}{3}$
⇒ 3(x + y) = xy
⇒ 3 × 16 = xy      [Since from (i), we have: x + y = 16]
∴ xy = 48                     ....(iii)
We know:
(x − y)² = (x + y)² − 4xy
(x − y)² = (16)² − 4 × 48 = 256 − 192 = 64
∴ (x − y) = $\pm \sqrt{64} = \pm 8$
Since x is larger and y is smaller, we have:
x − y = 8           .....(iv)
On adding (i) and (iv), we get:
2x = 24
⇒ x = 12
On substituting x = 12 in (i), we get:
12 + y = 16 ⇒ y = (16 − 12) = 4
Hence, the required numbers are 12 and 4.

Answer:

Let the number of students in classroom A be x
Let the number of students in classroom B be y.
If 10 students are transferred from A to B, then we have:
x − 10 = y + 10
⇒ x − y = 20 ...(i)
If 20 students are transferred from B to A, then we have:
2(y − 20) = x + 20
⇒ 2y − 40 = x + 20
⇒ −x + 2y = 60 ...(ii)
On adding (i) and (ii), we get:
y = (20 + 60) = 80
On substituting y = 80 in (i), we get:
x − 80 = 20
⇒ x = (20 + 80) = 100
Hence, the number of students in classroom A is 100 and the number of students in classroom B is 80.

Question 26:

Let the number of students in classroom A be x
Let the number of students in classroom B be y.
If 10 students are transferred from A to B, then we have:
x − 10 = y + 10
⇒ x − y = 20 ...(i)
If 20 students are transferred from B to A, then we have:
2(y − 20) = x + 20
⇒ 2y − 40 = x + 20
⇒ −x + 2y = 60 ...(ii)
On adding (i) and (ii), we get:
y = (20 + 60) = 80
On substituting y = 80 in (i), we get:
x − 80 = 20
⇒ x = (20 + 80) = 100
Hence, the number of students in classroom A is 100 and the number of students in classroom B is 80.

Answer:

Let fixed charges be â‚¹x and rate per km be â‚¹y.
Then as per the question
$x + 80 y = 1330 . . . . . (i) x + 90 y = 1490 . . . . . (i i)$
Subtracting (i) from (ii), we get
$10 y = 160 \Rightarrow y = \frac{160}{10} = 16$
Now, putting y = 16, we have
$x + 80 \times 16 = 1330 \Rightarrow x = 1330 - 1280 = 50$
Hence, the fixed charges be â‚¹50 and the rate per km is â‚¹16.

Question 27:

Let fixed charges be â‚¹x and rate per km be â‚¹y.
Then as per the question
$x + 80 y = 1330 . . . . . (i) x + 90 y = 1490 . . . . . (i i)$
Subtracting (i) from (ii), we get
$10 y = 160 \Rightarrow y = \frac{160}{10} = 16$
Now, putting y = 16, we have
$x + 80 \times 16 = 1330 \Rightarrow x = 1330 - 1280 = 50$
Hence, the fixed charges be â‚¹50 and the rate per km is â‚¹16.

Answer:

Let the fixed charges be â‚¹x and the cost of food per day be â‚¹y.
Then as per the question
$x + 25 y = 4500 . . . . . (i) x + 30 y = 5200 . . . . . (i i)$
Subtracting (i) from (ii), we get
$5 y = 700 \Rightarrow y = \frac{700}{5} = 140$
Now, putting y = 140, we have
$x + 25 \times 140 = 4500 \Rightarrow x = 4500 - 3500 = 1000$
Hence, the fixed charges is â‚¹1000 and the cost of the food per day is â‚¹140.

Question 28:

Let the fixed charges be â‚¹x and the cost of food per day be â‚¹y.
Then as per the question
$x + 25 y = 4500 . . . . . (i) x + 30 y = 5200 . . . . . (i i)$
Subtracting (i) from (ii), we get
$5 y = 700 \Rightarrow y = \frac{700}{5} = 140$
Now, putting y = 140, we have
$x + 25 \times 140 = 4500 \Rightarrow x = 4500 - 3500 = 1000$
Hence, the fixed charges is â‚¹1000 and the cost of the food per day is â‚¹140.

Answer:

Let the the amounts invested at 10% and 8% be â‚¹x and â‚¹y respectively.
Then as per the question
$\frac{x \times 10 \times 1}{100} + \frac{y \times 8 \times 1}{100} = 1350 10 x + 8 y = 135000 . . . . . (i)$
After the amounts interchanged but the rate being the same, we have
$\frac{x \times 8 \times 1}{100} + \frac{y \times 10 \times 1}{100} = 1350 - 45 8 x + 10 y = 130500 . . . . . (i i)$
Adding (i) and (ii) and dividing by 9, we get
$2 x + 2 y = 29500 . . . . . (i i i)$
Subtracting (ii) from (i), we get
$2 x - 2 y = 4500 . . . . . (i v)$
Now, adding (iii) and (iv), we have
$4 x = 34000 \Rightarrow x = \frac{34000}{4} = 8500$
Putting x = 8500 in (iii), we get
$2 \times 8500 + 2 y = 29500 \Rightarrow 2 y = 29500 - 17000 = 12500 \Rightarrow y = \frac{12500}{2} = 6250$
Hence, the amounts invested are â‚¹8,500 at 10% and â‚¹6,250 at 8%.

Question 29:

Let the the amounts invested at 10% and 8% be â‚¹x and â‚¹y respectively.
Then as per the question
$\frac{x \times 10 \times 1}{100} + \frac{y \times 8 \times 1}{100} = 1350 10 x + 8 y = 135000 . . . . . (i)$
After the amounts interchanged but the rate being the same, we have
$\frac{x \times 8 \times 1}{100} + \frac{y \times 10 \times 1}{100} = 1350 - 45 8 x + 10 y = 130500 . . . . . (i i)$
Adding (i) and (ii) and dividing by 9, we get
$2 x + 2 y = 29500 . . . . . (i i i)$
Subtracting (ii) from (i), we get
$2 x - 2 y = 4500 . . . . . (i v)$
Now, adding (iii) and (iv), we have
$4 x = 34000 \Rightarrow x = \frac{34000}{4} = 8500$
Putting x = 8500 in (iii), we get
$2 \times 8500 + 2 y = 29500 \Rightarrow 2 y = 29500 - 17000 = 12500 \Rightarrow y = \frac{12500}{2} = 6250$
Hence, the amounts invested are â‚¹8,500 at 10% and â‚¹6,250 at 8%.

Answer:

Let the monthly income of A and B are â‚¹x and â‚¹y respectively.
Then as per the question
$\frac{x}{y} = \frac{5}{4} \Rightarrow y = \frac{4 x}{5} . . . . . (i)$
Since each save â‚¹9,000, so
Expenditure of A = â€‹â‚¹ $(x - 9000)$
Expenditure of B = â€‹â‚¹ $(y - 9000)$
The ratio of expenditures of A and B are in the ratio 7 : 5.
$∴ \frac{x - 9000}{y - 9000} = \frac{7}{5} \Rightarrow 7 y - 63000 = 5 x - 45000 \Rightarrow 7 y - 5 x = 18000 . . . . . (i i)$
From (i), substitute $y = \frac{4 x}{5}$ in (ii) to get
$7 \times \frac{4 x}{5} - 5 x = 18000 \Rightarrow 28 x - 25 x = 90000 \Rightarrow 3 x = 90000 \Rightarrow x = 30000$
Now, putting x = 30000, we get
$y = \frac{4 \times 30000}{5} = 4 \times 6000 = 24000$
Hence, the monthly incomes of A and B are â€‹â‚¹30,000 and â€‹â‚¹24,000.

Question 30:

Let the monthly income of A and B are â‚¹x and â‚¹y respectively.
Then as per the question
$\frac{x}{y} = \frac{5}{4} \Rightarrow y = \frac{4 x}{5} . . . . . (i)$
Since each save â‚¹9,000, so
Expenditure of A = â€‹â‚¹ $(x - 9000)$
Expenditure of B = â€‹â‚¹ $(y - 9000)$
The ratio of expenditures of A and B are in the ratio 7 : 5.
$∴ \frac{x - 9000}{y - 9000} = \frac{7}{5} \Rightarrow 7 y - 63000 = 5 x - 45000 \Rightarrow 7 y - 5 x = 18000 . . . . . (i i)$
From (i), substitute $y = \frac{4 x}{5}$ in (ii) to get
$7 \times \frac{4 x}{5} - 5 x = 18000 \Rightarrow 28 x - 25 x = 90000 \Rightarrow 3 x = 90000 \Rightarrow x = 30000$
Now, putting x = 30000, we get
$y = \frac{4 \times 30000}{5} = 4 \times 6000 = 24000$
Hence, the monthly incomes of A and B are â€‹â‚¹30,000 and â€‹â‚¹24,000.

Answer:

Let the cost price of the chair and table be â‚¹x and â‚¹y respectively.
Then as per the question
Selling price of chair + Selling price of table = 1520
$\frac{100 + 25}{100} \times x + \frac{100 + 10}{100} \times y = 1520 \Rightarrow \frac{125}{100} x + \frac{110}{100} y = 1520 \Rightarrow 25 x + 22 y - 30400 = 0 . . . . . (i)$
When the profit on chair and table are 10% and 25% respectively, then

$\frac{100 + 10}{100} \times x + \frac{100 + 25}{100} \times y = 1535 \Rightarrow \frac{110}{100} x + \frac{125}{100} y = 1535 \Rightarrow 22 x + 25 y - 30700 = 0 . . . . . (i i)$
Solving (i) and (ii) by cross multiplication, we get
$\frac{x}{(22) (- 30700) - (25) (- 30400)} = \frac{y}{(- 30400) (22) - (- 30700) (25)} = \frac{1}{(25) (25) - (22) (22)} \Rightarrow \frac{x}{7600 - 6754} = \frac{y}{7675 - 6688} = \frac{100}{3 \times 47} \Rightarrow \frac{x}{846} = \frac{y}{987} = \frac{100}{3 \times 47} \Rightarrow x = \frac{100 \times 846}{3 \times 47}, y = \frac{100 \times 987}{3 \times 47}$
$\Rightarrow x = 600, y = 700$
Hence, the cost of chair and table are â€‹â‚¹600 and â€‹â‚¹700 respectively.

Question 31:

Let the cost price of the chair and table be â‚¹x and â‚¹y respectively.
Then as per the question
Selling price of chair + Selling price of table = 1520
$\frac{100 + 25}{100} \times x + \frac{100 + 10}{100} \times y = 1520 \Rightarrow \frac{125}{100} x + \frac{110}{100} y = 1520 \Rightarrow 25 x + 22 y - 30400 = 0 . . . . . (i)$
When the profit on chair and table are 10% and 25% respectively, then

$\frac{100 + 10}{100} \times x + \frac{100 + 25}{100} \times y = 1535 \Rightarrow \frac{110}{100} x + \frac{125}{100} y = 1535 \Rightarrow 22 x + 25 y - 30700 = 0 . . . . . (i i)$
Solving (i) and (ii) by cross multiplication, we get
$\frac{x}{(22) (- 30700) - (25) (- 30400)} = \frac{y}{(- 30400) (22) - (- 30700) (25)} = \frac{1}{(25) (25) - (22) (22)} \Rightarrow \frac{x}{7600 - 6754} = \frac{y}{7675 - 6688} = \frac{100}{3 \times 47} \Rightarrow \frac{x}{846} = \frac{y}{987} = \frac{100}{3 \times 47} \Rightarrow x = \frac{100 \times 846}{3 \times 47}, y = \frac{100 \times 987}{3 \times 47}$
$\Rightarrow x = 600, y = 700$
Hence, the cost of chair and table are â€‹â‚¹600 and â€‹â‚¹700 respectively.

Answer:

Let X and Y be the cars starting from points A and B, respectively and let their speeds be x km/h and y km/h, respectively.
Then, we have the following cases:
Case I: When the two cars move in the same direction
In this case, let the two cars meet at point M.

Distance covered by car X in 7 hours = 7x km
Distance covered by car Y in 7 hours = 7y km
∴ AM = (7x) km and BM = (7y) km
⇒ (AM − BM) = AB
⇒ (7x − 7y) = 70
⇒ 7(x − y) = 70
⇒ (x − y) = 10 ....(i)

Case II: When the two cars move in opposite directions
In this case, let the two cars meet at point N.

Distance covered by car X in 1 hour = x km
Distance covered by car Y in 1 hour = y km
∴ AN = x km and BN = y km
⇒ AN + BN = AB
⇒ x + y = 70 ....(ii)
On adding (i) and (ii), we get:
2x = 80
⇒ x = 40
On substituting x = 40 in (i), we get:
40 − y = 10
⇒ y = (40 − 10) = 30
Hence, the speed of car X is 40 km/h and the speed of car Y is 30 km/h.

Question 32:

Let X and Y be the cars starting from points A and B, respectively and let their speeds be x km/h and y km/h, respectively.
Then, we have the following cases:
Case I: When the two cars move in the same direction
In this case, let the two cars meet at point M.

Distance covered by car X in 7 hours = 7x km
Distance covered by car Y in 7 hours = 7y km
∴ AM = (7x) km and BM = (7y) km
⇒ (AM − BM) = AB
⇒ (7x − 7y) = 70
⇒ 7(x − y) = 70
⇒ (x − y) = 10 ....(i)

Case II: When the two cars move in opposite directions
In this case, let the two cars meet at point N.

Distance covered by car X in 1 hour = x km
Distance covered by car Y in 1 hour = y km
∴ AN = x km and BN = y km
⇒ AN + BN = AB
⇒ x + y = 70 ....(ii)
On adding (i) and (ii), we get:
2x = 80
⇒ x = 40
On substituting x = 40 in (i), we get:
40 − y = 10
⇒ y = (40 − 10) = 30
Hence, the speed of car X is 40 km/h and the speed of car Y is 30 km/h.

Answer:

Let the original speed be x kmph and let the time taken to complete the journey be y hours.
∴ Length of the whole journey = (xy) km
Case I:
When the speed is (x + 5) kmph and the time taken is (y − 3) hrs:
Total journey = (x + 5) (y − 3) km
⇒ (x + 5) (y − 3) = xy
⇒ xy + 5y − 3x − 15 = xy
⇒ 5y − 3x = 15 ...(i)
Case II:
When the speed is (x − 4) kmph and the time taken is (y + 3) hrs:
Total journey = (x − 4) (y + 3) km
⇒ (x − 4) (y + 3) = xy
⇒ xy − 4y + 3x − 12 = xy
⇒ 3x − 4y = 12 ...(ii)
On adding (i) and (ii), we get:
y = 27
On substituting y = 27 in (i), we get:
5 × 27 − 3x = 15
⇒ 135 − 3x = 15
⇒ 3x = 120
⇒ x = 40

∴ Length of the journey = (xy) km = (40 × 27) km = 1080 km

Question 33:

Let the original speed be x kmph and let the time taken to complete the journey be y hours.
∴ Length of the whole journey = (xy) km
Case I:
When the speed is (x + 5) kmph and the time taken is (y − 3) hrs:
Total journey = (x + 5) (y − 3) km
⇒ (x + 5) (y − 3) = xy
⇒ xy + 5y − 3x − 15 = xy
⇒ 5y − 3x = 15 ...(i)
Case II:
When the speed is (x − 4) kmph and the time taken is (y + 3) hrs:
Total journey = (x − 4) (y + 3) km
⇒ (x − 4) (y + 3) = xy
⇒ xy − 4y + 3x − 12 = xy
⇒ 3x − 4y = 12 ...(ii)
On adding (i) and (ii), we get:
y = 27
On substituting y = 27 in (i), we get:
5 × 27 − 3x = 15
⇒ 135 − 3x = 15
⇒ 3x = 120
⇒ x = 40

∴ Length of the journey = (xy) km = (40 × 27) km = 1080 km

Answer:

Let the speed of the train and taxi be x km/h and y km/h respectively.
Then as per the question
$\frac{3}{x} + \frac{2}{y} = \frac{11}{200} . . . . . (i)$
When the speeds of the train and taxi are 260 km and 240 km respectively, then
$\frac{260}{x} + \frac{240}{y} = \frac{11}{2} + \frac{6}{60} \Rightarrow \frac{13}{x} + \frac{12}{y} = \frac{28}{100} . . . . . (i i)$
Multiplying (i) by 6 and subtracting (ii) from it, we get
$\frac{18}{x} - \frac{13}{x} = \frac{66}{200} - \frac{28}{100} \Rightarrow \frac{5}{x} = \frac{10}{200} \Rightarrow x = 100$
Putting x = 100 in (i), we have
$\frac{3}{100} + \frac{2}{y} = \frac{11}{200} \Rightarrow \frac{2}{y} = \frac{11}{200} - \frac{3}{100} = \frac{1}{40} \Rightarrow y = 80$
Hence, the speed of the train and that of the taxi are 100 km/h and 80 km/h respectively.

Question 34:

Let the speed of the train and taxi be x km/h and y km/h respectively.
Then as per the question
$\frac{3}{x} + \frac{2}{y} = \frac{11}{200} . . . . . (i)$
When the speeds of the train and taxi are 260 km and 240 km respectively, then
$\frac{260}{x} + \frac{240}{y} = \frac{11}{2} + \frac{6}{60} \Rightarrow \frac{13}{x} + \frac{12}{y} = \frac{28}{100} . . . . . (i i)$
Multiplying (i) by 6 and subtracting (ii) from it, we get
$\frac{18}{x} - \frac{13}{x} = \frac{66}{200} - \frac{28}{100} \Rightarrow \frac{5}{x} = \frac{10}{200} \Rightarrow x = 100$
Putting x = 100 in (i), we have
$\frac{3}{100} + \frac{2}{y} = \frac{11}{200} \Rightarrow \frac{2}{y} = \frac{11}{200} - \frac{3}{100} = \frac{1}{40} \Rightarrow y = 80$
Hence, the speed of the train and that of the taxi are 100 km/h and 80 km/h respectively.

Answer:

Let the speed of the car A and B be x km/h and y km/h respectively . Let x > y.
Case-1: When they travel in the same direction

From the figure
$A C - B C = 160 \Rightarrow x \times 8 - y \times 8 = 160 \Rightarrow x - y = 20 . . . . . (i)$
Case-2: When they travel in opposite direction

From the figure
$A C + B C = 160 \Rightarrow x \times 2 + y \times 2 = 160 \Rightarrow x + y = 80 . . . . . (i i)$
Adding (i) and (ii), we get
$2 x = 100 \Rightarrow x = 50 km / h$
Putting x = 50 in (ii), we have
$50 + y = 80 \Rightarrow y = 80 - 50 = 30 km / h$
Hence, the speeds of the cars are 50 km/h and 30 km/h.

Question 35:

Let the speed of the car A and B be x km/h and y km/h respectively . Let x > y.
Case-1: When they travel in the same direction

From the figure
$A C - B C = 160 \Rightarrow x \times 8 - y \times 8 = 160 \Rightarrow x - y = 20 . . . . . (i)$
Case-2: When they travel in opposite direction

From the figure
$A C + B C = 160 \Rightarrow x \times 2 + y \times 2 = 160 \Rightarrow x + y = 80 . . . . . (i i)$
Adding (i) and (ii), we get
$2 x = 100 \Rightarrow x = 50 km / h$
Putting x = 50 in (ii), we have
$50 + y = 80 \Rightarrow y = 80 - 50 = 30 km / h$
Hence, the speeds of the cars are 50 km/h and 30 km/h.

Answer:

Let the speed of the sailor in still water be x km/h and that of the current y km/h.
Speed downstream = (x + y) km/h
Speed upstream = (x − y) km/h
As per the question
$(x + y) \times \frac{40}{60} = 8 \Rightarrow x + y = 12 . . . . . (i)$
When the sailor goes upstream, then
$(x - y) \times 1 = 8 x - y = 8 . . . . . (i i)$
Adding (i) and (ii), we get
$2 x = 20 \Rightarrow x = 10$
Putting x = 10 in (i), we have
$10 + y = 12 \Rightarrow y = 2$
Hence, the speeds of the sailor in staill water and the current are 10 km/h and 2 km/h respectively.

Question 36:

Let the speed of the sailor in still water be x km/h and that of the current y km/h.
Speed downstream = (x + y) km/h
Speed upstream = (x − y) km/h
As per the question
$(x + y) \times \frac{40}{60} = 8 \Rightarrow x + y = 12 . . . . . (i)$
When the sailor goes upstream, then
$(x - y) \times 1 = 8 x - y = 8 . . . . . (i i)$
Adding (i) and (ii), we get
$2 x = 20 \Rightarrow x = 10$
Putting x = 10 in (i), we have
$10 + y = 12 \Rightarrow y = 2$
Hence, the speeds of the sailor in staill water and the current are 10 km/h and 2 km/h respectively.

Answer:

Let speed of boat in still water be x km/h and speed of stream be y km/h.
Speed Upstream = (x − y) km/h
Speed downstream = (x + y) km/h

According to the question,
$\frac{30}{x - y} + \frac{44}{x + y} = 10 and \frac{40}{x - y} + \frac{55}{x + y} = 13 Let \frac{1}{x - y} = p and \frac{1}{x + y} = q . . . (1) T h e r e f o r e, t h e e q u a t i o n b e c o m e s 30 p + 44 q = 10 . . . (2) 40 p + 55 q = 13 . . . (3) 40 p + 55 q = 13 \Rightarrow 40 p = 13 - 55 q \Rightarrow p = \frac{13 - 55 q}{40} . . . (4) Substituting the value of p in (2), we get 30 (\frac{13 - 55 q}{40}) + 44 q = 10 \Rightarrow 30 (\frac{13 - 55 q}{40}) + 44 q = 10 \Rightarrow \frac{3 (13 - 55 q) + 4 (44 q)}{4} = 10 \Rightarrow 39 - 165 q + 176 q = 40 \Rightarrow 11 q = 40 - 39 \Rightarrow 11 q = 1 \Rightarrow q = \frac{1}{11} . . . (5) Substituting the value of q in (4), we get p = \frac{13 - 55 \frac{1}{11}}{40} \Rightarrow p = \frac{13 - 5}{40} \Rightarrow p = \frac{8}{40} \Rightarrow p = \frac{1}{5} . . . (6) From (1), (5) and (6), we get x - y = 5 and x + y = 11 Solving both, we get x = 8 and y = 3$

Hence, the speed of the stream and that of the boat in still water is 3 km/h and 8 km/h, respectively.

Question 37:

Let speed of boat in still water be x km/h and speed of stream be y km/h.
Speed Upstream = (x − y) km/h
Speed downstream = (x + y) km/h

According to the question,
$\frac{30}{x - y} + \frac{44}{x + y} = 10 and \frac{40}{x - y} + \frac{55}{x + y} = 13 Let \frac{1}{x - y} = p and \frac{1}{x + y} = q . . . (1) T h e r e f o r e, t h e e q u a t i o n b e c o m e s 30 p + 44 q = 10 . . . (2) 40 p + 55 q = 13 . . . (3) 40 p + 55 q = 13 \Rightarrow 40 p = 13 - 55 q \Rightarrow p = \frac{13 - 55 q}{40} . . . (4) Substituting the value of p in (2), we get 30 (\frac{13 - 55 q}{40}) + 44 q = 10 \Rightarrow 30 (\frac{13 - 55 q}{40}) + 44 q = 10 \Rightarrow \frac{3 (13 - 55 q) + 4 (44 q)}{4} = 10 \Rightarrow 39 - 165 q + 176 q = 40 \Rightarrow 11 q = 40 - 39 \Rightarrow 11 q = 1 \Rightarrow q = \frac{1}{11} . . . (5) Substituting the value of q in (4), we get p = \frac{13 - 55 \frac{1}{11}}{40} \Rightarrow p = \frac{13 - 5}{40} \Rightarrow p = \frac{8}{40} \Rightarrow p = \frac{1}{5} . . . (6) From (1), (5) and (6), we get x - y = 5 and x + y = 11 Solving both, we get x = 8 and y = 3$

Hence, the speed of the stream and that of the boat in still water is 3 km/h and 8 km/h, respectively.

Answer:

Let us suppose that one man alone can finish the work in x days and one boy alone can finish it in y days.
∴ One man's one day's work = $\frac{1}{x}$
And, one boy's one day's work = $\frac{1}{y}$
2 men and 5 boys can finish the work in 4 days.
∴ (2 men's one day's work) + (5 boys' one day's work) = $\frac{1}{4}$
⇒ $\frac{2}{x} + \frac{5}{y} = \frac{1}{4}$
⇒ $2 u + 5 v = \frac{1}{4}$               ...(i)           Here, $\frac{1}{x} = u and \frac{1}{y} = v$
Again, 3 men and 6 boys can finish the work in 3 days.
∴ (3 men's one day's work) + (6 boys' one day's work) = $\frac{1}{3}$
⇒ $\frac{3}{x} + \frac{6}{y} = \frac{1}{3}$
⇒ $3 u + 6 v = \frac{1}{3}$              ....(ii)           Here, $\frac{1}{x} = u and \frac{1}{y} = v$
On multiplying (i) by 6 and (ii) by 5, we get:
$12 u + 30 v = \frac{6}{4}$              ....(iii)
$15 u + 30 v = \frac{5}{3}$              ....(iv)
On subtracting (iii) from (iv), we get:
$3 u = (\frac{5}{3} - \frac{6}{4}) = \frac{2}{12} = \frac{1}{6}$
⇒ $u = \frac{1}{6 \times 3} = \frac{1}{18} \Rightarrow \frac{1}{x} = \frac{1}{18} \Rightarrow x = 18$
On substituting $u = \frac{1}{18}$ in (i), we get:
$2 \times \frac{1}{18} + 5 v = \frac{1}{4} \Rightarrow 5 v = (\frac{1}{4} - \frac{1}{9}) = \frac{5}{36}$
⇒ $v = (\frac{5}{36} \times \frac{1}{5}) = \frac{1}{36} \Rightarrow \frac{1}{y} = \frac{1}{36} \Rightarrow y = 36$
Hence, one man alone can finish the work in 18 days and one boy alone can finish the work in 36 days.

Question 38:

Let us suppose that one man alone can finish the work in x days and one boy alone can finish it in y days.
∴ One man's one day's work = $\frac{1}{x}$
And, one boy's one day's work = $\frac{1}{y}$
2 men and 5 boys can finish the work in 4 days.
∴ (2 men's one day's work) + (5 boys' one day's work) = $\frac{1}{4}$
⇒ $\frac{2}{x} + \frac{5}{y} = \frac{1}{4}$
⇒ $2 u + 5 v = \frac{1}{4}$               ...(i)           Here, $\frac{1}{x} = u and \frac{1}{y} = v$
Again, 3 men and 6 boys can finish the work in 3 days.
∴ (3 men's one day's work) + (6 boys' one day's work) = $\frac{1}{3}$
⇒ $\frac{3}{x} + \frac{6}{y} = \frac{1}{3}$
⇒ $3 u + 6 v = \frac{1}{3}$              ....(ii)           Here, $\frac{1}{x} = u and \frac{1}{y} = v$
On multiplying (i) by 6 and (ii) by 5, we get:
$12 u + 30 v = \frac{6}{4}$              ....(iii)
$15 u + 30 v = \frac{5}{3}$              ....(iv)
On subtracting (iii) from (iv), we get:
$3 u = (\frac{5}{3} - \frac{6}{4}) = \frac{2}{12} = \frac{1}{6}$
⇒ $u = \frac{1}{6 \times 3} = \frac{1}{18} \Rightarrow \frac{1}{x} = \frac{1}{18} \Rightarrow x = 18$
On substituting $u = \frac{1}{18}$ in (i), we get:
$2 \times \frac{1}{18} + 5 v = \frac{1}{4} \Rightarrow 5 v = (\frac{1}{4} - \frac{1}{9}) = \frac{5}{36}$
⇒ $v = (\frac{5}{36} \times \frac{1}{5}) = \frac{1}{36} \Rightarrow \frac{1}{y} = \frac{1}{36} \Rightarrow y = 36$
Hence, one man alone can finish the work in 18 days and one boy alone can finish the work in 36 days.

Answer:

Let the length of the room be x metres and the breadth of the room be y metres.
Then, we have:
Area of the room = xy
According to the question, we have:
x = y + 3
⇒ x − y = 3           ....(i)
And, (x + 3) (y − 2) = xy
⇒ xy − 2x + 3y − 6 = xy
⇒ 3y − 2x = 6           ....(ii)
On multiplying (i) by 2, we get:
2x − 2y = 6           ....(iii)
On adding (ii) and (iii), we get:
y = (6 + 6) = 12
On substituting y = 12 in (i), we get:
x − 12 = 3
⇒ x = (3 + 12) =15
Hence, the length of the room is 15 metres and its breadth is 12 metres.

Question 39:

Let the length of the room be x metres and the breadth of the room be y metres.
Then, we have:
Area of the room = xy
According to the question, we have:
x = y + 3
⇒ x − y = 3           ....(i)
And, (x + 3) (y − 2) = xy
⇒ xy − 2x + 3y − 6 = xy
⇒ 3y − 2x = 6           ....(ii)
On multiplying (i) by 2, we get:
2x − 2y = 6           ....(iii)
On adding (ii) and (iii), we get:
y = (6 + 6) = 12
On substituting y = 12 in (i), we get:
x − 12 = 3
⇒ x = (3 + 12) =15
Hence, the length of the room is 15 metres and its breadth is 12 metres.

Answer:

Let the length and the breadth of the rectangle be x m and y m, respectively.
∴ Area of the rectangle = (xy) sq. m

Case 1: When the length is reduced by 5 m and the breadth is increased by 3 m:
New length = (x − 5) m
New breadth = (y + 3) m
∴ New area = (x − 5) (y + 3) sq. m
∴ xy − (x − 5) (y + 3) = 8
⇒ xy − [xy − 5y + 3x − 15] = 8
⇒ xy − xy + 5y − 3x + 15 = 8
⇒ 3x − 5y = 7                .....(i)

Case 2: When the length is increased by 3 m and the breadth is increased by 2 m:
New length = (x + 3) m
New breadth = (y + 2) m
∴ New area = (x + 3) (y + 2) sq. m
∴ (x + 3) (y + 2) − xy = 74
⇒ [xy + 3y + 2x + 6] − xy = 74
⇒ 2x + 3y = 68             .....(ii)
On multiplying (i) by 3 and (ii) by 5, we get:
9x − 15y = 21              .....(iii)
10x + 15y = 340          .....(iv)
On adding (iii) and (iv), we get:
19x = 361
⇒ x = 19
On substituting x = 19 in (iii), we get:
9 × 19 − 15y = 21
⇒ 171 −15y = 21
⇒ 15y = (171 − 21) = 150
⇒ y = 10
Hence, the length is 19 m and the breadth is 10 m.

Question 40:

Let the length and the breadth of the rectangle be x m and y m, respectively.
∴ Area of the rectangle = (xy) sq. m

Case 1: When the length is reduced by 5 m and the breadth is increased by 3 m:
New length = (x − 5) m
New breadth = (y + 3) m
∴ New area = (x − 5) (y + 3) sq. m
∴ xy − (x − 5) (y + 3) = 8
⇒ xy − [xy − 5y + 3x − 15] = 8
⇒ xy − xy + 5y − 3x + 15 = 8
⇒ 3x − 5y = 7                .....(i)

Case 2: When the length is increased by 3 m and the breadth is increased by 2 m:
New length = (x + 3) m
New breadth = (y + 2) m
∴ New area = (x + 3) (y + 2) sq. m
∴ (x + 3) (y + 2) − xy = 74
⇒ [xy + 3y + 2x + 6] − xy = 74
⇒ 2x + 3y = 68             .....(ii)
On multiplying (i) by 3 and (ii) by 5, we get:
9x − 15y = 21              .....(iii)
10x + 15y = 340          .....(iv)
On adding (iii) and (iv), we get:
19x = 361
⇒ x = 19
On substituting x = 19 in (iii), we get:
9 × 19 − 15y = 21
⇒ 171 −15y = 21
⇒ 15y = (171 − 21) = 150
⇒ y = 10
Hence, the length is 19 m and the breadth is 10 m.

Answer:

Let the length and breadth of the rectangle be x m and y m respectively.
Case1: When length is increased by 3 m and breadth is decreased by 4 m
$x y - (x + 3) (y - 4) = 67 \Rightarrow x y - x y + 4 x - 3 y + 12 = 67 \Rightarrow 4 x - 3 y = 55 . . . . . (i)$
Case2: When length is reduced by 1 m and breadth is increased by 4 m
$(x - 1) (y + 4) - x y = 89 \Rightarrow x y + 4 x - y - 4 - x y = 89 \Rightarrow 4 x - y = 93 . . . . . (i i)$
Subtracting (i) from (ii), we get
$2 y = 38 \Rightarrow y = 19$
Now, putting y = 19 in (ii), we have
$4 x - 19 = 93 \Rightarrow 4 x = 93 + 19 = 112 \Rightarrow x = 28$
Hence, length = 28 m and breadth = 19 m.

Question 41:

Let the length and breadth of the rectangle be x m and y m respectively.
Case1: When length is increased by 3 m and breadth is decreased by 4 m
$x y - (x + 3) (y - 4) = 67 \Rightarrow x y - x y + 4 x - 3 y + 12 = 67 \Rightarrow 4 x - 3 y = 55 . . . . . (i)$
Case2: When length is reduced by 1 m and breadth is increased by 4 m
$(x - 1) (y + 4) - x y = 89 \Rightarrow x y + 4 x - y - 4 - x y = 89 \Rightarrow 4 x - y = 93 . . . . . (i i)$
Subtracting (i) from (ii), we get
$2 y = 38 \Rightarrow y = 19$
Now, putting y = 19 in (ii), we have
$4 x - 19 = 93 \Rightarrow 4 x = 93 + 19 = 112 \Rightarrow x = 28$
Hence, length = 28 m and breadth = 19 m.

Answer:

Let the the basic first class full fare be â‚¹x and the reservation charge be â‚¹y.
Case 1: One reservation first class full ticket cost â‚¹4,150
$x + y = 4150 . . . . . (i)$
Case 2: One full and one half reserved first class tickets cost â‚¹6,255
$(x + y) + (\frac{1}{2} x + y) = 6255 \Rightarrow 3 x + 4 y = 12510 . . . . . (i i)$
Substituting $y = 4150 - x$ from (i) in (ii), we get
$3 x + 4 (4150 - x) = 12510 \Rightarrow 3 x - 4 x + 16600 = 12510 \Rightarrow x = 16600 - 12510 = 4090$
Now, putting x = 4090 in (i), we have
$4090 + y = 4150 \Rightarrow y = 4150 - 4090 = 60$
Hence, cost of basic first class full fare = â‚¹4,090 and reservation charge = â‚¹60.

Question 42:

Let the the basic first class full fare be â‚¹x and the reservation charge be â‚¹y.
Case 1: One reservation first class full ticket cost â‚¹4,150
$x + y = 4150 . . . . . (i)$
Case 2: One full and one half reserved first class tickets cost â‚¹6,255
$(x + y) + (\frac{1}{2} x + y) = 6255 \Rightarrow 3 x + 4 y = 12510 . . . . . (i i)$
Substituting $y = 4150 - x$ from (i) in (ii), we get
$3 x + 4 (4150 - x) = 12510 \Rightarrow 3 x - 4 x + 16600 = 12510 \Rightarrow x = 16600 - 12510 = 4090$
Now, putting x = 4090 in (i), we have
$4090 + y = 4150 \Rightarrow y = 4150 - 4090 = 60$
Hence, cost of basic first class full fare = â‚¹4,090 and reservation charge = â‚¹60.

Answer:

Let the the present age of the man be x years and that of his son be y years.
After 5 years man's age = x + 5
After 5 years ago son's age = y + 5
As per the question
$x + 5 = 3 (y + 5) \Rightarrow x - 3 y = 10 . . . . . (i)$
5 years ago man's age = x − 5
5 years ago son's age = y − 5
As per the question
$x - 5 = 7 (y - 5) \Rightarrow x - 7 y = - 30 . . . . . (i i)$
Subtracting (ii) from (i), we have
$4 y = 40 \Rightarrow y = 10$
Putting y = 10 in (i), we get
$x - 3 \times 10 = 10 \Rightarrow x = 10 + 30 = 40$
Hence, man's present age = 40 years and son's present age = 10 years.

Question 43:

Let the the present age of the man be x years and that of his son be y years.
After 5 years man's age = x + 5
After 5 years ago son's age = y + 5
As per the question
$x + 5 = 3 (y + 5) \Rightarrow x - 3 y = 10 . . . . . (i)$
5 years ago man's age = x − 5
5 years ago son's age = y − 5
As per the question
$x - 5 = 7 (y - 5) \Rightarrow x - 7 y = - 30 . . . . . (i i)$
Subtracting (ii) from (i), we have
$4 y = 40 \Rightarrow y = 10$
Putting y = 10 in (i), we get
$x - 3 \times 10 = 10 \Rightarrow x = 10 + 30 = 40$
Hence, man's present age = 40 years and son's present age = 10 years.

Answer:

Let the man's present age be x years.
Let his son's present age be y years.
According to question, we have:
Two years ago:
Age of the man = Five times the age of the son
⇒ (x − 2) = 5(y − 2)
⇒ x − 2 = 5y − 10
⇒ x − 5y = −8 .....(i)
Two years later:
Age of the man = Three times the age of the son + 8
⇒ (x + 2) = 3(y + 2) + 8
⇒ x + 2 = 3y + 6 + 8
⇒ x − 3y = 12 .....(ii)
Subtracting (i) from (ii), we get:
2y = 20
⇒ y = 10
On substituting y = 10 in (i), we get:
x − 5 × 10 = −8
⇒ x − 50 = −8
⇒ x = (−8 + 50) = 42
Hence, the present age of the man is 42 years and the present age of the son is 10 years.

Question 44:

Let the man's present age be x years.
Let his son's present age be y years.
According to question, we have:
Two years ago:
Age of the man = Five times the age of the son
⇒ (x − 2) = 5(y − 2)
⇒ x − 2 = 5y − 10
⇒ x − 5y = −8 .....(i)
Two years later:
Age of the man = Three times the age of the son + 8
⇒ (x + 2) = 3(y + 2) + 8
⇒ x + 2 = 3y + 6 + 8
⇒ x − 3y = 12 .....(ii)
Subtracting (i) from (ii), we get:
2y = 20
⇒ y = 10
On substituting y = 10 in (i), we get:
x − 5 × 10 = −8
⇒ x − 50 = −8
⇒ x = (−8 + 50) = 42
Hence, the present age of the man is 42 years and the present age of the son is 10 years.

Answer:

Let the father's present age be x years.
Let her son's present age be y years.
Then, we have:
x + 2y = 70                    ....(i)
And, 2x + y = 95               ....(ii)
On multiplying (ii) by 2, we get:
4x + 2y = 190                 ....(iii)
On subtracting (i) from (iii), we get:
3x = 120
⇒ x = 40
On substituting x = 40 in (i), we get:
40 + 2y = 70
⇒ 2y = (70 − 40) = 30
⇒ y = 15
Hence, the father's present age is 40 years and her son's present age is 15 years.

Question 45:

Let the father's present age be x years.
Let her son's present age be y years.
Then, we have:
x + 2y = 70                    ....(i)
And, 2x + y = 95               ....(ii)
On multiplying (ii) by 2, we get:
4x + 2y = 190                 ....(iii)
On subtracting (i) from (iii), we get:
3x = 120
⇒ x = 40
On substituting x = 40 in (i), we get:
40 + 2y = 70
⇒ 2y = (70 − 40) = 30
⇒ y = 15
Hence, the father's present age is 40 years and her son's present age is 15 years.

Answer:

Let the woman's present age be x years.
Let her daughter's present age be y years.
Then, we have:
x = 3y + 3
⇒ x − 3y = 3 ....(i)
After three years, we have:
(x + 3) = 2(y + 3) + 10
⇒ x + 3 = 2y + 6 + 10
⇒ x − 2y = 13 ....(ii)
Subtracting (ii) from (i), we get:
−y = (3 − 13) = −10
⇒ y = 10
On substituting y = 10 in (i), we get:
x − 3 × 10 = 3
⇒ x − 30 = 3
⇒ x = (3 + 30) = 33
Hence, the woman's present age is 33 years and her daughter's present age is 10 years.

Question 46:

Let the woman's present age be x years.
Let her daughter's present age be y years.
Then, we have:
x = 3y + 3
⇒ x − 3y = 3 ....(i)
After three years, we have:
(x + 3) = 2(y + 3) + 10
⇒ x + 3 = 2y + 6 + 10
⇒ x − 2y = 13 ....(ii)
Subtracting (ii) from (i), we get:
−y = (3 − 13) = −10
⇒ y = 10
On substituting y = 10 in (i), we get:
x − 3 × 10 = 3
⇒ x − 30 = 3
⇒ x = (3 + 30) = 33
Hence, the woman's present age is 33 years and her daughter's present age is 10 years.

Answer:

Let the actual price of the tea and lemon set be â‚¹x and â‚¹y respectively.
When gain is â‚¹7, then
$\frac{y}{100} \times 15 - \frac{x}{100} \times 5 = 7 \Rightarrow 3 y - x = 140 . . . . . (i)$
When gain is â‚¹14, then
$\frac{y}{100} \times 5 + \frac{x}{100} \times 10 = 14 \Rightarrow y + 2 x = 280 . . . . . (i i)$
Multiplying (i) by 2 and adding with (ii), we have
$7 y = 280 + 280 \Rightarrow y = \frac{560}{7} = 80$
Putting y = 80 in (ii), we get
$80 + 2 x = 280 \Rightarrow x = \frac{200}{2} = 100$
Hence, actual price of the tea set and lemon set are â‚¹100 and â‚¹80 respectively.

Question 47:

Let the actual price of the tea and lemon set be â‚¹x and â‚¹y respectively.
When gain is â‚¹7, then
$\frac{y}{100} \times 15 - \frac{x}{100} \times 5 = 7 \Rightarrow 3 y - x = 140 . . . . . (i)$
When gain is â‚¹14, then
$\frac{y}{100} \times 5 + \frac{x}{100} \times 10 = 14 \Rightarrow y + 2 x = 280 . . . . . (i i)$
Multiplying (i) by 2 and adding with (ii), we have
$7 y = 280 + 280 \Rightarrow y = \frac{560}{7} = 80$
Putting y = 80 in (ii), we get
$80 + 2 x = 280 \Rightarrow x = \frac{200}{2} = 100$
Hence, actual price of the tea set and lemon set are â‚¹100 and â‚¹80 respectively.

Answer:

Let the fixed charge be â‚¹x and the charge for each extra day be â‚¹y.
In case of Mona, as per the question
$x + 4 y = 27 . . . . . (i)$
In case of Tanvy, as per the question
$x + 2 y = 21 . . . . . (i i)$
Subtracting (ii) from (i), we get
$2 y = 6 \Rightarrow y = 3$
Now, putting y = 3 in (ii), we have
$x + 2 \times 3 = 21 \Rightarrow x = 21 - 6 = 15$
Hence, the fixed charge be â‚¹15 and the charge for each extra day is â‚¹3.

Question 48:

Let the fixed charge be â‚¹x and the charge for each extra day be â‚¹y.
In case of Mona, as per the question
$x + 4 y = 27 . . . . . (i)$
In case of Tanvy, as per the question
$x + 2 y = 21 . . . . . (i i)$
Subtracting (ii) from (i), we get
$2 y = 6 \Rightarrow y = 3$
Now, putting y = 3 in (ii), we have
$x + 2 \times 3 = 21 \Rightarrow x = 21 - 6 = 15$
Hence, the fixed charge be â‚¹15 and the charge for each extra day is â‚¹3.

Answer:

Let the digit at the tens place be x and digit at the units place be y.
The number is 10x + y.

According to the question,
$x + y = 9 . . . (1) 10 x + y + 27 = 10 y + x . . . (2) 10 x + y + 27 = 10 y + x \Rightarrow 10 x + y - 10 y - x = - 27 \Rightarrow 9 x - 9 y = - 27 \Rightarrow 9 (x - y) = - 27 \Rightarrow x - y = - 3 \Rightarrow x = - 3 + y . . . (3) Substituting the value of x in (1), we get - 3 + y + y = 9 \Rightarrow 2 y = 9 + 3 \Rightarrow 2 y = 12 \Rightarrow y = 6 . . . (4) Substituting the value of y in (3), we get x = - 3 + 6 \Rightarrow x = 3 Hence, the number is 36 .$

Question 49:

Let the digit at the tens place be x and digit at the units place be y.
The number is 10x + y.

According to the question,
$x + y = 9 . . . (1) 10 x + y + 27 = 10 y + x . . . (2) 10 x + y + 27 = 10 y + x \Rightarrow 10 x + y - 10 y - x = - 27 \Rightarrow 9 x - 9 y = - 27 \Rightarrow 9 (x - y) = - 27 \Rightarrow x - y = - 3 \Rightarrow x = - 3 + y . . . (3) Substituting the value of x in (1), we get - 3 + y + y = 9 \Rightarrow 2 y = 9 + 3 \Rightarrow 2 y = 12 \Rightarrow y = 6 . . . (4) Substituting the value of y in (3), we get x = - 3 + 6 \Rightarrow x = 3 Hence, the number is 36 .$

Answer:

Let the numerator of the fraction be x and the denominator be y.

According to the question,

$\frac{x - 2}{y} = \frac{1}{3} . . . (i) and \frac{x}{y - 1} = \frac{1}{2} . . . (ii) Solving equation (i), we get 3 (x - 2) = y \Rightarrow 3 x - 6 = y . . . (iii) Substituting the value of y in equation (ii), we get \frac{x}{3 x - 6 - 1} = \frac{1}{2} \Rightarrow 2 (x) = 3 x - 6 - 1 \Rightarrow 2 x = 3 x - 7 \Rightarrow 2 x - 3 x = - 7 \Rightarrow - x = - 7 \Rightarrow x = 7 . . . (iv) From (iii) and (iv), we get x = 7 and y = 15 Hence, the fraction is \frac{7}{15} .$

Question 50:

Let the numerator of the fraction be x and the denominator be y.

According to the question,

$\frac{x - 2}{y} = \frac{1}{3} . . . (i) and \frac{x}{y - 1} = \frac{1}{2} . . . (ii) Solving equation (i), we get 3 (x - 2) = y \Rightarrow 3 x - 6 = y . . . (iii) Substituting the value of y in equation (ii), we get \frac{x}{3 x - 6 - 1} = \frac{1}{2} \Rightarrow 2 (x) = 3 x - 6 - 1 \Rightarrow 2 x = 3 x - 7 \Rightarrow 2 x - 3 x = - 7 \Rightarrow - x = - 7 \Rightarrow x = 7 . . . (iv) From (iii) and (iv), we get x = 7 and y = 15 Hence, the fraction is \frac{7}{15} .$

Answer:

Let the age of first son be x and of second son be y.
Then, the father's present age be 3(x + y).

According to the question,
After 5 years, the father's age will be two times the sum of the ages of his two children.

$3 (x + y) + 5 = 2 (x + 5 + y + 5) \Rightarrow 3 x + 3 y + 5 = 2 x + 10 + 2 y + 10 \Rightarrow 3 x + 3 y - 2 x - 2 y = 20 - 5 \Rightarrow x + y = 15 \Rightarrow 3 (x + y) = 45$

Hence, the present age of the father is 45 years.

Question 51:

Let the age of first son be x and of second son be y.
Then, the father's present age be 3(x + y).

According to the question,
After 5 years, the father's age will be two times the sum of the ages of his two children.

$3 (x + y) + 5 = 2 (x + 5 + y + 5) \Rightarrow 3 x + 3 y + 5 = 2 x + 10 + 2 y + 10 \Rightarrow 3 x + 3 y - 2 x - 2 y = 20 - 5 \Rightarrow x + y = 15 \Rightarrow 3 (x + y) = 45$

Hence, the present age of the father is 45 years.

Answer:

Let the length of the side of one square be x m and the length of the side of another square be y m.

According to the question,
$x^{2} + y^{2} = 640 . . . (1) 4 x - 4 y = 64 . . . (2) 4 x - 4 y = 64 \Rightarrow 4 (x - y) = 64 \Rightarrow x - y = 16 \Rightarrow x = 16 + y . . . (3) Substituting the value of x in (1), we get {(16 + y)}^{2} + y^{2} = 640 \Rightarrow 256 + y^{2} + 32 y + y^{2} = 640 \Rightarrow 2 y^{2} + 32 y + 256 - 640 = 0 \Rightarrow 2 y^{2} + 32 y - 384 = 0 \Rightarrow 2 (y^{2} + 16 y - 192) = 0 \Rightarrow y^{2} + 16 y - 192 = 0 \Rightarrow y^{2} + 24 y - 8 y - 192 = 0 \Rightarrow y (y + 24) - 8 (y + 24) = 0 \Rightarrow (y - 8) (y + 24) = 0 \Rightarrow y = 8 . . . (4) (∵ y > 0) Substituting the value of y in (3), we get x = 16 + 8 \Rightarrow x = 24 Hence, the sides of the squares are 24 m and 8 m .$

Question 52:

Let the length of the side of one square be x m and the length of the side of another square be y m.

According to the question,
$x^{2} + y^{2} = 640 . . . (1) 4 x - 4 y = 64 . . . (2) 4 x - 4 y = 64 \Rightarrow 4 (x - y) = 64 \Rightarrow x - y = 16 \Rightarrow x = 16 + y . . . (3) Substituting the value of x in (1), we get {(16 + y)}^{2} + y^{2} = 640 \Rightarrow 256 + y^{2} + 32 y + y^{2} = 640 \Rightarrow 2 y^{2} + 32 y + 256 - 640 = 0 \Rightarrow 2 y^{2} + 32 y - 384 = 0 \Rightarrow 2 (y^{2} + 16 y - 192) = 0 \Rightarrow y^{2} + 16 y - 192 = 0 \Rightarrow y^{2} + 24 y - 8 y - 192 = 0 \Rightarrow y (y + 24) - 8 (y + 24) = 0 \Rightarrow (y - 8) (y + 24) = 0 \Rightarrow y = 8 . . . (4) (∵ y > 0) Substituting the value of y in (3), we get x = 16 + 8 \Rightarrow x = 24 Hence, the sides of the squares are 24 m and 8 m .$

Answer:

Let the length of the side of one square be x m and the length of the side of another square be y m.

According to the question,
$x^{2} + y^{2} = 157 . . . (1) 4 x + 4 y = 68 . . . (2) 4 x + 4 y = 68 \Rightarrow 4 (x + y) = 68 \Rightarrow x + y = 17 \Rightarrow x = 17 - y . . . (3) Substituting the value of x in (1), we get {(17 - y)}^{2} + y^{2} = 157 \Rightarrow 289 + y^{2} - 34 y + y^{2} = 157 \Rightarrow 2 y^{2} - 34 y + 289 - 157 = 0 \Rightarrow 2 y^{2} - 34 y + 132 = 0 \Rightarrow 2 (y^{2} - 17 y + 66) = 0 \Rightarrow y^{2} - 17 y + 66 = 0 \Rightarrow y^{2} - 11 y - 6 y + 66 = 0 \Rightarrow y (y - 11) - 6 (y - 11) = 0 \Rightarrow (y - 6) (y - 11) = 0 \Rightarrow y = 6, 11 . . . (4) (∵ y > 0) Substituting the value of y in (3), we get x = 17 - 6 and x = 17 - 11 \Rightarrow x = 11, 6 Hence, the sides of the squares are 11 m and 6 m .$

Question 53:

Let the length of the side of one square be x m and the length of the side of another square be y m.

According to the question,
$x^{2} + y^{2} = 157 . . . (1) 4 x + 4 y = 68 . . . (2) 4 x + 4 y = 68 \Rightarrow 4 (x + y) = 68 \Rightarrow x + y = 17 \Rightarrow x = 17 - y . . . (3) Substituting the value of x in (1), we get {(17 - y)}^{2} + y^{2} = 157 \Rightarrow 289 + y^{2} - 34 y + y^{2} = 157 \Rightarrow 2 y^{2} - 34 y + 289 - 157 = 0 \Rightarrow 2 y^{2} - 34 y + 132 = 0 \Rightarrow 2 (y^{2} - 17 y + 66) = 0 \Rightarrow y^{2} - 17 y + 66 = 0 \Rightarrow y^{2} - 11 y - 6 y + 66 = 0 \Rightarrow y (y - 11) - 6 (y - 11) = 0 \Rightarrow (y - 6) (y - 11) = 0 \Rightarrow y = 6, 11 . . . (4) (∵ y > 0) Substituting the value of y in (3), we get x = 17 - 6 and x = 17 - 11 \Rightarrow x = 11, 6 Hence, the sides of the squares are 11 m and 6 m .$

Answer:

Let x litres and y litres be the amount of acids from 50% and 25% acid solutions respectively.
As per the question
$50 % of x + 25 % of y = 40 % of 10 \Rightarrow 0.50 x + 0.25 y = 4 \Rightarrow 2 x + y = 16 . . . . . (i)$
Since, the total volume is 10 litres, so
$x + y = 10 . . . . . (i i)$
Subtracting (ii) from (i), we get
$x = 6$
Now, putting x = 6 in (ii), we have
$6 + y = 10 \Rightarrow y = 4$
Hence, volume of 50% acid solution = 6 litres and volume of 25% acid solution = 4 litres.

Question 54:

Let x litres and y litres be the amount of acids from 50% and 25% acid solutions respectively.
As per the question
$50 % of x + 25 % of y = 40 % of 10 \Rightarrow 0.50 x + 0.25 y = 4 \Rightarrow 2 x + y = 16 . . . . . (i)$
Since, the total volume is 10 litres, so
$x + y = 10 . . . . . (i i)$
Subtracting (ii) from (i), we get
$x = 6$
Now, putting x = 6 in (ii), we have
$6 + y = 10 \Rightarrow y = 4$
Hence, volume of 50% acid solution = 6 litres and volume of 25% acid solution = 4 litres.

Answer:

Let x g and y g be the weight of 18-carat and 12-carat gold respectively.
As per the given condition
$\frac{18 x}{24} + \frac{12 y}{24} = \frac{120 \times 16}{24} \Rightarrow 3 x + 2 y = 320 . . . . . (i)$
And
$x + y = 120 . . . . . (i i)$
Multiplying (ii) by 2 and subtracting from (i), we get
$x = 320 - 240 = 80$
Now, putting x = 80 in (ii), we have
$80 + y = 120 \Rightarrow y = 40$
Hence, the required weight of 18-carat and 12-carat gold bars are 80 g and 40 g respectively.

Question 55:

Let x g and y g be the weight of 18-carat and 12-carat gold respectively.
As per the given condition
$\frac{18 x}{24} + \frac{12 y}{24} = \frac{120 \times 16}{24} \Rightarrow 3 x + 2 y = 320 . . . . . (i)$
And
$x + y = 120 . . . . . (i i)$
Multiplying (ii) by 2 and subtracting from (i), we get
$x = 320 - 240 = 80$
Now, putting x = 80 in (ii), we have
$80 + y = 120 \Rightarrow y = 40$
Hence, the required weight of 18-carat and 12-carat gold bars are 80 g and 40 g respectively.

Answer:

Let x litres and y litres be respectively the amount of 90% and 97% pure acid solutions.
As per the given condition
$0.90 x + 0.97 y = 21 \times 0.95 \Rightarrow 0.90 x + 0.97 y = 21 \times 0.95 . . . . . (i)$
And
$x + y = 21 . . . . . (i i)$
From (ii), subtitute $y = 21 - x$ in (i) to get
$0.90 x + 0.97 (21 - x) = 21 \times 0.95 \Rightarrow 0.90 x + 0.97 \times 21 - 0.97 x = 21 \times 0.95 \Rightarrow 0.07 x = 0.97 \times 21 - 21 \times 0.95 \Rightarrow x = \frac{21 \times 0.02}{0.07} = 6$
Now, putting x = 6 in (ii), we have
$6 + y = 21 \Rightarrow y = 15$
Hence, the required quantities are 6 litres and 15 litres.

Question 56:

Let x litres and y litres be respectively the amount of 90% and 97% pure acid solutions.
As per the given condition
$0.90 x + 0.97 y = 21 \times 0.95 \Rightarrow 0.90 x + 0.97 y = 21 \times 0.95 . . . . . (i)$
And
$x + y = 21 . . . . . (i i)$
From (ii), subtitute $y = 21 - x$ in (i) to get
$0.90 x + 0.97 (21 - x) = 21 \times 0.95 \Rightarrow 0.90 x + 0.97 \times 21 - 0.97 x = 21 \times 0.95 \Rightarrow 0.07 x = 0.97 \times 21 - 21 \times 0.95 \Rightarrow x = \frac{21 \times 0.02}{0.07} = 6$
Now, putting x = 6 in (ii), we have
$6 + y = 21 \Rightarrow y = 15$
Hence, the required quantities are 6 litres and 15 litres.

Answer:

Let x and y be the supplementary angles, where x > y.
As per the given condition
$x + y = 180^{\circ} . . . . . (i)$
And
$x - y = 18^{\circ} . . . . . (i i)$
Adding (i) and (ii), we get
$2 x = 198^{\circ} \Rightarrow x = 99^{\circ}$
Now, substituting $x = 99^{\circ}$ in (ii), we have
$99^{\circ} - y = 18^{\circ} \Rightarrow x = 99^{\circ} - 18^{\circ} = 81^{\circ}$
Hence, the required angles are $99^{\circ}$ and $81^{\circ}$ .

Question 57:

Let x and y be the supplementary angles, where x > y.
As per the given condition
$x + y = 180^{\circ} . . . . . (i)$
And
$x - y = 18^{\circ} . . . . . (i i)$
Adding (i) and (ii), we get
$2 x = 198^{\circ} \Rightarrow x = 99^{\circ}$
Now, substituting $x = 99^{\circ}$ in (ii), we have
$99^{\circ} - y = 18^{\circ} \Rightarrow x = 99^{\circ} - 18^{\circ} = 81^{\circ}$
Hence, the required angles are $99^{\circ}$ and $81^{\circ}$ .

Answer:

$∵ ∠ C - ∠ B = 9^{\circ} ∴ y^{\circ} - {(3 x - 2)}^{\circ} = 9^{\circ} \Rightarrow y^{\circ} - 3 x^{\circ} + 2^{\circ} = 9^{\circ} \Rightarrow y^{\circ} - 3 x^{\circ} = 7^{\circ} . . . . . (i)$
The sum of all the angles of a triangle is $180^{\circ}$ , therefore
$∠ A + ∠ B + ∠ C = 180^{\circ} \Rightarrow x^{\circ} + {(3 x - 2)}^{\circ} + y^{\circ} = 180^{\circ} \Rightarrow 4 x^{\circ} + y^{\circ} = 182^{\circ} . . . . . (i i)$
Subtracting (i) from (ii), we have
$7 x^{\circ} = 182^{\circ} - 7^{\circ} = 175^{\circ} \Rightarrow x^{\circ} = 25^{\circ}$
Now, substituting $x^{\circ} = 25^{\circ}$ in (i), we have
$y^{\circ} = 3 x^{\circ} + 7^{\circ} = 3 \times 25^{\circ} + 7^{\circ} = 82^{\circ}$
Thus
$∠ A = x^{\circ} = 25^{\circ} ∠ B = {(3 x - 2)}^{\circ} = 75^{\circ} - 2^{\circ} = 73^{\circ} ∠ C = y^{\circ} = 82^{\circ}$
Hence, the angles are $25^{\circ}, 73^{\circ} and 82^{\circ}$ .

Question 58:

$∵ ∠ C - ∠ B = 9^{\circ} ∴ y^{\circ} - {(3 x - 2)}^{\circ} = 9^{\circ} \Rightarrow y^{\circ} - 3 x^{\circ} + 2^{\circ} = 9^{\circ} \Rightarrow y^{\circ} - 3 x^{\circ} = 7^{\circ} . . . . . (i)$
The sum of all the angles of a triangle is $180^{\circ}$ , therefore
$∠ A + ∠ B + ∠ C = 180^{\circ} \Rightarrow x^{\circ} + {(3 x - 2)}^{\circ} + y^{\circ} = 180^{\circ} \Rightarrow 4 x^{\circ} + y^{\circ} = 182^{\circ} . . . . . (i i)$
Subtracting (i) from (ii), we have
$7 x^{\circ} = 182^{\circ} - 7^{\circ} = 175^{\circ} \Rightarrow x^{\circ} = 25^{\circ}$
Now, substituting $x^{\circ} = 25^{\circ}$ in (i), we have
$y^{\circ} = 3 x^{\circ} + 7^{\circ} = 3 \times 25^{\circ} + 7^{\circ} = 82^{\circ}$
Thus
$∠ A = x^{\circ} = 25^{\circ} ∠ B = {(3 x - 2)}^{\circ} = 75^{\circ} - 2^{\circ} = 73^{\circ} ∠ C = y^{\circ} = 82^{\circ}$
Hence, the angles are $25^{\circ}, 73^{\circ} and 82^{\circ}$ .

Answer:

The opposite angles of cyclic quadrilateral are supplementary, so
$∠ A + ∠ C = 180^{\circ} \Rightarrow {(2 x + 4)}^{\circ} + {(2 y + 10)}^{\circ} = 180^{\circ} \Rightarrow x + y = 83^{\circ} . . . . . (i)$
And
$∠ B + ∠ D = 180^{\circ} \Rightarrow {(y + 3)}^{\circ} + {(4 x - 5)}^{\circ} = 180^{\circ} \Rightarrow 4 x + y = 182^{\circ} . . . . . (i i)$
Subtracting (i) from (ii), we have
$3 x = 99 \Rightarrow x = 33^{\circ}$
Now, substituting $x = 33^{\circ}$ in (i), we have
$33^{\circ} + y = 83^{\circ} \Rightarrow y = 83^{\circ} - 33^{\circ} = 50^{\circ}$
Therefore
$∠ A = {(2 x + 4)}^{\circ} = {(2 \times 33 + 4)}^{\circ} = 70^{\circ} ∠ B = {(y + 3)}^{\circ} = {(50 + 3)}^{\circ} = 53^{\circ} ∠ C = {(2 y + 10)}^{\circ} = {(2 \times 50 + 10)}^{\circ} = 110^{\circ} ∠ D = {(4 x - 5)}^{\circ} = {(4 \times 33 - 5)}^{\circ} = 132 ° - 5^{\circ} = 127^{\circ}$
Hence, $∠ A = 70^{\circ}, ∠ B = 53^{\circ}, ∠ C = 110^{\circ} and ∠ D = 127^{\circ}$ .

Question 1:

The opposite angles of cyclic quadrilateral are supplementary, so
$∠ A + ∠ C = 180^{\circ} \Rightarrow {(2 x + 4)}^{\circ} + {(2 y + 10)}^{\circ} = 180^{\circ} \Rightarrow x + y = 83^{\circ} . . . . . (i)$
And
$∠ B + ∠ D = 180^{\circ} \Rightarrow {(y + 3)}^{\circ} + {(4 x - 5)}^{\circ} = 180^{\circ} \Rightarrow 4 x + y = 182^{\circ} . . . . . (i i)$
Subtracting (i) from (ii), we have
$3 x = 99 \Rightarrow x = 33^{\circ}$
Now, substituting $x = 33^{\circ}$ in (i), we have
$33^{\circ} + y = 83^{\circ} \Rightarrow y = 83^{\circ} - 33^{\circ} = 50^{\circ}$
Therefore
$∠ A = {(2 x + 4)}^{\circ} = {(2 \times 33 + 4)}^{\circ} = 70^{\circ} ∠ B = {(y + 3)}^{\circ} = {(50 + 3)}^{\circ} = 53^{\circ} ∠ C = {(2 y + 10)}^{\circ} = {(2 \times 50 + 10)}^{\circ} = 110^{\circ} ∠ D = {(4 x - 5)}^{\circ} = {(4 \times 33 - 5)}^{\circ} = 132 ° - 5^{\circ} = 127^{\circ}$
Hence, $∠ A = 70^{\circ}, ∠ B = 53^{\circ}, ∠ C = 110^{\circ} and ∠ D = 127^{\circ}$ .

Answer:

The given equations are
$x + 2 y - 8 = 0 . . . . . (i) 2 x + 4 y - 16 = 0 . . . . . (i i)$
Which is of the form $a_{1} x + b_{1} y + c_{1} = 0 and a_{2} x + b_{2} y + c_{2} = 0$ , where $a_{1} = 1, b_{1} = 2, c_{1} = - 8, a_{2} = 2, b_{2} = 4 and c_{2} = - 18$
Now
$\frac{a_{1}}{a_{2}} = \frac{1}{2} \frac{b_{1}}{b_{2}} = \frac{2}{4} = \frac{1}{2} \frac{c_{1}}{c_{2}} = \frac{- 8}{- 16} = \frac{1}{2}$
$\Rightarrow \frac{a_{1}}{a_{2}} = \frac{b_{1}}{b_{2}} = \frac{c_{1}}{c_{2}} = \frac{1}{2}$
Thus, the pair of linear equations are coincident and therefore has infinitely many solutions.

Question 2:

The given equations are
$x + 2 y - 8 = 0 . . . . . (i) 2 x + 4 y - 16 = 0 . . . . . (i i)$
Which is of the form $a_{1} x + b_{1} y + c_{1} = 0 and a_{2} x + b_{2} y + c_{2} = 0$ , where $a_{1} = 1, b_{1} = 2, c_{1} = - 8, a_{2} = 2, b_{2} = 4 and c_{2} = - 18$
Now
$\frac{a_{1}}{a_{2}} = \frac{1}{2} \frac{b_{1}}{b_{2}} = \frac{2}{4} = \frac{1}{2} \frac{c_{1}}{c_{2}} = \frac{- 8}{- 16} = \frac{1}{2}$
$\Rightarrow \frac{a_{1}}{a_{2}} = \frac{b_{1}}{b_{2}} = \frac{c_{1}}{c_{2}} = \frac{1}{2}$
Thus, the pair of linear equations are coincident and therefore has infinitely many solutions.

Answer:

The given equations are
$2 x + 3 y - 7 = 0 . . . . . (i) (k - 1) x + (k + 2) y - 3 k = 0 . . . . . (i i)$
Which is of the form $a_{1} x + b_{1} y + c_{1} = 0 and a_{2} x + b_{2} y + c_{2} = 0$ , where $a_{1} = 2, b_{1} = 3, c_{1} = - 7, a_{2} = k - 1, b_{2} = k + 2 and c_{2} = - 3 k$
For the given pair of linear equations to have infinitely many solutions, we must have
$\frac{a_{1}}{a_{2}} = \frac{b_{1}}{b_{2}} = \frac{c_{1}}{c_{2}} \Rightarrow \frac{2}{k - 1} = \frac{3}{k + 2} = \frac{- 7}{- 3 k} \Rightarrow \frac{2}{k - 1} = \frac{3}{k + 2}, \frac{3}{k + 2} = \frac{- 7}{- 3 k} and \frac{2}{k - 1} = \frac{- 7}{- 3 k} \Rightarrow 2 (k + 2) = 3 (k - 1), 9 k = 7 k + 14 and 6 k = 7 k - 7$
$\Rightarrow k = 7, k = 7 and k = 7$
Hence, k = 7.

Question 3:

The given equations are
$2 x + 3 y - 7 = 0 . . . . . (i) (k - 1) x + (k + 2) y - 3 k = 0 . . . . . (i i)$
Which is of the form $a_{1} x + b_{1} y + c_{1} = 0 and a_{2} x + b_{2} y + c_{2} = 0$ , where $a_{1} = 2, b_{1} = 3, c_{1} = - 7, a_{2} = k - 1, b_{2} = k + 2 and c_{2} = - 3 k$
For the given pair of linear equations to have infinitely many solutions, we must have
$\frac{a_{1}}{a_{2}} = \frac{b_{1}}{b_{2}} = \frac{c_{1}}{c_{2}} \Rightarrow \frac{2}{k - 1} = \frac{3}{k + 2} = \frac{- 7}{- 3 k} \Rightarrow \frac{2}{k - 1} = \frac{3}{k + 2}, \frac{3}{k + 2} = \frac{- 7}{- 3 k} and \frac{2}{k - 1} = \frac{- 7}{- 3 k} \Rightarrow 2 (k + 2) = 3 (k - 1), 9 k = 7 k + 14 and 6 k = 7 k - 7$
$\Rightarrow k = 7, k = 7 and k = 7$
Hence, k = 7.

Answer:

The given pair of linear equations are
$10 x + 5 y - (k - 5) = 0 . . . . . (i) 20 x + 10 y - k = 0 . . . . . (i i)$
Which is of the form $a_{1} x + b_{1} y + c_{1} = 0 and a_{2} x + b_{2} y + c_{2} = 0$ , where $a_{1} = 10, b_{1} = 5, c_{1} = - (k - 5), a_{2} = 20, b_{2} = 10 and c_{2} = - k$
For the given pair of linear equations to have infinitely many solutions, we must have
$\frac{a_{1}}{a_{2}} = \frac{b_{1}}{b_{2}} = \frac{c_{1}}{c_{2}} \Rightarrow \frac{10}{20} = \frac{5}{10} = \frac{- (k - 5)}{- k} \Rightarrow \frac{1}{2} = \frac{k - 5}{k} \Rightarrow 2 k - 10 = k \Rightarrow k = 10$
Hence, k = 10.

Question 4:

The given pair of linear equations are
$10 x + 5 y - (k - 5) = 0 . . . . . (i) 20 x + 10 y - k = 0 . . . . . (i i)$
Which is of the form $a_{1} x + b_{1} y + c_{1} = 0 and a_{2} x + b_{2} y + c_{2} = 0$ , where $a_{1} = 10, b_{1} = 5, c_{1} = - (k - 5), a_{2} = 20, b_{2} = 10 and c_{2} = - k$
For the given pair of linear equations to have infinitely many solutions, we must have
$\frac{a_{1}}{a_{2}} = \frac{b_{1}}{b_{2}} = \frac{c_{1}}{c_{2}} \Rightarrow \frac{10}{20} = \frac{5}{10} = \frac{- (k - 5)}{- k} \Rightarrow \frac{1}{2} = \frac{k - 5}{k} \Rightarrow 2 k - 10 = k \Rightarrow k = 10$
Hence, k = 10.

Answer:

The given pair of linear equations are
$2 x + 3 y - 9 = 0 . . . . . (i) 6 x + (k - 2) y - (3 k - 2) = 0 . . . . . (i i)$
Which is of the form $a_{1} x + b_{1} y + c_{1} = 0 and a_{2} x + b_{2} y + c_{2} = 0$ , where $a_{1} = 2, b_{1} = 3, c_{1} = - 9, a_{2} = 6, b_{2} = k - 2 and c_{2} = - (3 k - 2)$
For the given pair of linear equations to have no solution, we must have
$\frac{a_{1}}{a_{2}} = \frac{b_{1}}{b_{2}} \neq \frac{c_{1}}{c_{2}} \Rightarrow \frac{2}{6} = \frac{3}{k - 2} \neq \frac{- 9}{- (3 k - 2)} \Rightarrow \frac{2}{6} = \frac{3}{k - 2}, \frac{3}{k - 2} \neq \frac{- 9}{- (3 k - 2)} \Rightarrow k = 11, \frac{3}{k - 2} \neq \frac{9}{3 k - 2}$
$\Rightarrow k = 11, 3 (3 k - 2) \neq 9 (k - 2) \Rightarrow k = 11, 1 \neq 3 (true)$
Hence, k = 11.

Question 5:

The given pair of linear equations are
$2 x + 3 y - 9 = 0 . . . . . (i) 6 x + (k - 2) y - (3 k - 2) = 0 . . . . . (i i)$
Which is of the form $a_{1} x + b_{1} y + c_{1} = 0 and a_{2} x + b_{2} y + c_{2} = 0$ , where $a_{1} = 2, b_{1} = 3, c_{1} = - 9, a_{2} = 6, b_{2} = k - 2 and c_{2} = - (3 k - 2)$
For the given pair of linear equations to have no solution, we must have
$\frac{a_{1}}{a_{2}} = \frac{b_{1}}{b_{2}} \neq \frac{c_{1}}{c_{2}} \Rightarrow \frac{2}{6} = \frac{3}{k - 2} \neq \frac{- 9}{- (3 k - 2)} \Rightarrow \frac{2}{6} = \frac{3}{k - 2}, \frac{3}{k - 2} \neq \frac{- 9}{- (3 k - 2)} \Rightarrow k = 11, \frac{3}{k - 2} \neq \frac{9}{3 k - 2}$
$\Rightarrow k = 11, 3 (3 k - 2) \neq 9 (k - 2) \Rightarrow k = 11, 1 \neq 3 (true)$
Hence, k = 11.

Answer:

The given pair of linear equations are
$x + 3 y - 4 = 0 . . . . . (i) 2 x + 6 y - 7 = 0 . . . . . (i i)$
Which is of the form $a_{1} x + b_{1} y + c_{1} = 0 and a_{2} x + b_{2} y + c_{2} = 0$ , where $a_{1} = 1, b_{1} = 3, c_{1} = - 4, a_{2} = 2, b_{2} = 6 and c_{2} = - 7$
Now
$\frac{a_{1}}{a_{2}} = \frac{1}{2} \frac{b_{1}}{b_{2}} = \frac{3}{6} = \frac{1}{2} \frac{c_{1}}{c_{2}} = \frac{- 4}{- 7} = \frac{4}{7} \Rightarrow \frac{a_{1}}{a_{2}} = \frac{b_{1}}{b_{2}} \neq \frac{c_{1}}{c_{2}}$
Thus, the pair of the given linear equations has no solution.

Question 6:

The given pair of linear equations are
$x + 3 y - 4 = 0 . . . . . (i) 2 x + 6 y - 7 = 0 . . . . . (i i)$
Which is of the form $a_{1} x + b_{1} y + c_{1} = 0 and a_{2} x + b_{2} y + c_{2} = 0$ , where $a_{1} = 1, b_{1} = 3, c_{1} = - 4, a_{2} = 2, b_{2} = 6 and c_{2} = - 7$
Now
$\frac{a_{1}}{a_{2}} = \frac{1}{2} \frac{b_{1}}{b_{2}} = \frac{3}{6} = \frac{1}{2} \frac{c_{1}}{c_{2}} = \frac{- 4}{- 7} = \frac{4}{7} \Rightarrow \frac{a_{1}}{a_{2}} = \frac{b_{1}}{b_{2}} \neq \frac{c_{1}}{c_{2}}$
Thus, the pair of the given linear equations has no solution.

Answer:

The given pair of linear equations is
$3 x + k y = 0 . . . . . (i) 2 x - y = 0 . . . . . (i i)$
Which is of the form $a_{1} x + b_{1} y + c_{1} = 0 and a_{2} x + b_{2} y + c_{2} = 0$ , where $a_{1} = 3, b_{1} = k, c_{1} = 0, a_{2} = 2, b_{2} = - 1 and c_{2} = 0$
For the system to have a unique solution, we must have
$\frac{a_{1}}{a_{2}} = \frac{b_{1}}{b_{2}} \Rightarrow \frac{3}{2} \neq \frac{k}{- 1} \Rightarrow k \neq - \frac{3}{2}$
Hence, $k \neq - \frac{3}{2}$ .

Question 7:

The given pair of linear equations is
$3 x + k y = 0 . . . . . (i) 2 x - y = 0 . . . . . (i i)$
Which is of the form $a_{1} x + b_{1} y + c_{1} = 0 and a_{2} x + b_{2} y + c_{2} = 0$ , where $a_{1} = 3, b_{1} = k, c_{1} = 0, a_{2} = 2, b_{2} = - 1 and c_{2} = 0$
For the system to have a unique solution, we must have
$\frac{a_{1}}{a_{2}} = \frac{b_{1}}{b_{2}} \Rightarrow \frac{3}{2} \neq \frac{k}{- 1} \Rightarrow k \neq - \frac{3}{2}$
Hence, $k \neq - \frac{3}{2}$ .

Answer:

Let the numbers be x and y, where x > y.
Then as per the question
$x - y = 5 . . . . . (i) x^{2} - y^{2} = 65 . . . . . (i i)$
Dividing (ii) by (i), we get
$\frac{x^{2} - y^{2}}{x - y} = \frac{65}{5} \Rightarrow \frac{(x - y) (x + y)}{x - y} = 13 \Rightarrow x + y = 13 . . . . . (i i i)$
Now, adding (i) and (ii), we have
$2 x = 18 \Rightarrow x = 9$
Substituting x = 9 in (iii), we have
$9 + y = 13 \Rightarrow y = 4$
Hence, the numbers are 9 and 4.

Question 8:

Let the numbers be x and y, where x > y.
Then as per the question
$x - y = 5 . . . . . (i) x^{2} - y^{2} = 65 . . . . . (i i)$
Dividing (ii) by (i), we get
$\frac{x^{2} - y^{2}}{x - y} = \frac{65}{5} \Rightarrow \frac{(x - y) (x + y)}{x - y} = 13 \Rightarrow x + y = 13 . . . . . (i i i)$
Now, adding (i) and (ii), we have
$2 x = 18 \Rightarrow x = 9$
Substituting x = 9 in (iii), we have
$9 + y = 13 \Rightarrow y = 4$
Hence, the numbers are 9 and 4.

Answer:

Let the cost of 1 pen and 1 pencil are â‚¹x and â‚¹y respectively.
Then as per the question
$5 x + 8 y = 120 . . . . . (i) 8 x + 5 y = 153 . . . . . (i i)$
Adding (i) and (ii), we get
$13 x + 13 y = 273 \Rightarrow x + y = 21 . . . . . (i i i)$
Subtracting (i) from (ii), we get
$3 x - 3 y = 33 \Rightarrow x - y = 11 . . . . . (i v)$
Now, adding (iii) and (iv), we get
$2 x = 32 \Rightarrow x = 16$
Substituting x = 16 in (iii), we have
$16 + y = 21 \Rightarrow y = 5$
Hence, the cost of 1 pen and 1 pencil are respectively â‚¹16 and â‚¹5.

Question 9:

Let the cost of 1 pen and 1 pencil are â‚¹x and â‚¹y respectively.
Then as per the question
$5 x + 8 y = 120 . . . . . (i) 8 x + 5 y = 153 . . . . . (i i)$
Adding (i) and (ii), we get
$13 x + 13 y = 273 \Rightarrow x + y = 21 . . . . . (i i i)$
Subtracting (i) from (ii), we get
$3 x - 3 y = 33 \Rightarrow x - y = 11 . . . . . (i v)$
Now, adding (iii) and (iv), we get
$2 x = 32 \Rightarrow x = 16$
Substituting x = 16 in (iii), we have
$16 + y = 21 \Rightarrow y = 5$
Hence, the cost of 1 pen and 1 pencil are respectively â‚¹16 and â‚¹5.

Answer:

Let the larger number be x and the smaller number be y.
Then as per the question
$x + y = 80 . . . . . (i) x = 4 y + 5 x - 4 y = 5 . . . . . (i i)$
Subtracting (ii) from (i), we get
$5 y = 75 \Rightarrow y = 15$
Now, putting y = 15 in (i), we have
$x + 15 = 80 \Rightarrow x = 65$
Hence, the numbers are 65 and 15.

Question 10:

Let the larger number be x and the smaller number be y.
Then as per the question
$x + y = 80 . . . . . (i) x = 4 y + 5 x - 4 y = 5 . . . . . (i i)$
Subtracting (ii) from (i), we get
$5 y = 75 \Rightarrow y = 15$
Now, putting y = 15 in (i), we have
$x + 15 = 80 \Rightarrow x = 65$
Hence, the numbers are 65 and 15.

Answer:

Let the ones digit and tens digit be x and y respectively.
Then as per the question
$x + y = 10 . . . . . (i) (10 y + x) - 18 = 10 x + y x - y = - 2 . . . . . (i i)$
Adding (i) and (ii), we get
$2 x = 8 \Rightarrow x = 4$
Now, putting x = 4 in (i), we have
$4 + y = 10 \Rightarrow y = 6$
Hence, the number is 64.

Question 11:

Let the ones digit and tens digit be x and y respectively.
Then as per the question
$x + y = 10 . . . . . (i) (10 y + x) - 18 = 10 x + y x - y = - 2 . . . . . (i i)$
Adding (i) and (ii), we get
$2 x = 8 \Rightarrow x = 4$
Now, putting x = 4 in (i), we have
$4 + y = 10 \Rightarrow y = 6$
Hence, the number is 64.

Answer:

Let the number of stamps of 20 p and 25 p be x and y respectively.
As per the question
$x + y = 47 . . . . . (i) 0.20 x + 0.25 y = 10 4 x + 5 y = 200 . . . . . (i i)$
From (i), we get
$y = 47 - x$
Now, substituting $y = 47 - x$ in (ii), we have
$4 x + 5 (47 - x) = 200 \Rightarrow 4 x - 5 x + 235 = 200 \Rightarrow x = 235 - 200 = 35$
Putting x = 35 in (i), we get
$35 + y = 47 \Rightarrow y = 47 - 35 = 12$
Hence, the number of 20 p stamps and 25 p stamps are 35 and 12 respectively.

Question 12:

Let the number of stamps of 20 p and 25 p be x and y respectively.
As per the question
$x + y = 47 . . . . . (i) 0.20 x + 0.25 y = 10 4 x + 5 y = 200 . . . . . (i i)$
From (i), we get
$y = 47 - x$
Now, substituting $y = 47 - x$ in (ii), we have
$4 x + 5 (47 - x) = 200 \Rightarrow 4 x - 5 x + 235 = 200 \Rightarrow x = 235 - 200 = 35$
Putting x = 35 in (i), we get
$35 + y = 47 \Rightarrow y = 47 - 35 = 12$
Hence, the number of 20 p stamps and 25 p stamps are 35 and 12 respectively.

Answer:

Let the number of hens and cow be x and y respectively.
As per the question
$x + y = 48 . . . . . (i) 2 x + 4 y = 140 x + 2 y = 70 . . . . . (i i)$
Subtracting (i) from (ii), we have
$y = 22$
Hence, the number of cows is 22.

Question 13:

Let the number of hens and cow be x and y respectively.
As per the question
$x + y = 48 . . . . . (i) 2 x + 4 y = 140 x + 2 y = 70 . . . . . (i i)$
Subtracting (i) from (ii), we have
$y = 22$
Hence, the number of cows is 22.

Answer:

The given pair of equation is
$\frac{2}{x} + \frac{3}{y} = \frac{9}{x y} . . . . . (i) \frac{4}{x} + \frac{9}{y} = \frac{21}{x y} . . . . . (i i)$
Multiplying (i) and (ii) by xy, we have
$3 x + 2 y = 9 . . . . . (i i i) 9 x + 4 y = 21 . . . . . (i v)$
Now, multiplying (iii) by 2 and subtracting from (iv), we get
$9 x - 6 x = 21 - 18 \Rightarrow x = \frac{3}{3} = 1$
Putting x = 1 in (iii), we have
$3 \times 1 + 2 y = 9 \Rightarrow y = \frac{9 - 3}{2} = 3$
Hence, x = 1 and y = 3.

Question 14:

The given pair of equation is
$\frac{2}{x} + \frac{3}{y} = \frac{9}{x y} . . . . . (i) \frac{4}{x} + \frac{9}{y} = \frac{21}{x y} . . . . . (i i)$
Multiplying (i) and (ii) by xy, we have
$3 x + 2 y = 9 . . . . . (i i i) 9 x + 4 y = 21 . . . . . (i v)$
Now, multiplying (iii) by 2 and subtracting from (iv), we get
$9 x - 6 x = 21 - 18 \Rightarrow x = \frac{3}{3} = 1$
Putting x = 1 in (iii), we have
$3 \times 1 + 2 y = 9 \Rightarrow y = \frac{9 - 3}{2} = 3$
Hence, x = 1 and y = 3.

Answer:

The given pair of equations is
$\frac{x}{4} + \frac{y}{3} = \frac{5}{12} . . . . . (i) \frac{x}{2} + y = 1 . . . . . (i i)$
Multiplying (i) by 12 and (ii) by 4, we get
$3 x + 4 y = 5 . . . . . (i i i) 2 x + 4 y = 4 . . . . . (i v)$
Now, subtracting (iv) from (iii), we get
$x = 1$
Putting x = 1 in (iv), we have
$2 + 4 y = 4 \Rightarrow 4 y = 2 \Rightarrow y = \frac{1}{2}$
$∴ x + y = 1 + \frac{1}{2} = \frac{3}{2}$
Hence, the value of x + y is $\frac{3}{2}$ .

Question 15:

The given pair of equations is
$\frac{x}{4} + \frac{y}{3} = \frac{5}{12} . . . . . (i) \frac{x}{2} + y = 1 . . . . . (i i)$
Multiplying (i) by 12 and (ii) by 4, we get
$3 x + 4 y = 5 . . . . . (i i i) 2 x + 4 y = 4 . . . . . (i v)$
Now, subtracting (iv) from (iii), we get
$x = 1$
Putting x = 1 in (iv), we have
$2 + 4 y = 4 \Rightarrow 4 y = 2 \Rightarrow y = \frac{1}{2}$
$∴ x + y = 1 + \frac{1}{2} = \frac{3}{2}$
Hence, the value of x + y is $\frac{3}{2}$ .

Answer:

The given pair of equations is
$12 x + 17 y = 53 . . . . . (i) 17 x + 12 y = 63 . . . . . (i i)$
Adding (i) and (ii), we get
$29 x + 29 y = 116 \Rightarrow x + y = 4 (Dividing by 4)$
Hence, the value of x + y is 4.

Question 16:

The given pair of equations is
$12 x + 17 y = 53 . . . . . (i) 17 x + 12 y = 63 . . . . . (i i)$
Adding (i) and (ii), we get
$29 x + 29 y = 116 \Rightarrow x + y = 4 (Dividing by 4)$
Hence, the value of x + y is 4.

Answer:

The given system is
$3 x + 5 y = 0 . . . . . (i) k x + 10 y = 0 . . . . . (i i)$
This is a homogeneous system of linear differential equation, so it always has a zero
solution i.e., x = y = 0.
But to have a nonzero solution, it must have infinitely many solutions.
For this, we have
$\frac{a_{1}}{a_{2}} = \frac{b_{1}}{b_{2}} \Rightarrow \frac{3}{k} = \frac{5}{10} = \frac{1}{2} \Rightarrow k = 6$
Hence, k = 6.

Question 17:

The given system is
$3 x + 5 y = 0 . . . . . (i) k x + 10 y = 0 . . . . . (i i)$
This is a homogeneous system of linear differential equation, so it always has a zero
solution i.e., x = y = 0.
But to have a nonzero solution, it must have infinitely many solutions.
For this, we have
$\frac{a_{1}}{a_{2}} = \frac{b_{1}}{b_{2}} \Rightarrow \frac{3}{k} = \frac{5}{10} = \frac{1}{2} \Rightarrow k = 6$
Hence, k = 6.

Answer:

The given system is
$k x - y - 2 = 0 . . . . . (i) 6 x - 2 y - 3 = 0 . . . . . (i i)$
Here, $a_{1} = k, b_{1} = - 1, c_{1} = - 2 . a_{2} = 6, b_{2} = - 2 and c_{2} = - 3$ .
For the system, to have a unique solution, we must have
$\frac{a_{1}}{a_{2}} \neq \frac{b_{1}}{b_{2}} \Rightarrow \frac{k}{6} \neq \frac{- 1}{- 2} = \frac{1}{2} \Rightarrow k \neq 3$
Hence, $k \neq 3$ .

Question 18:

The given system is
$k x - y - 2 = 0 . . . . . (i) 6 x - 2 y - 3 = 0 . . . . . (i i)$
Here, $a_{1} = k, b_{1} = - 1, c_{1} = - 2 . a_{2} = 6, b_{2} = - 2 and c_{2} = - 3$ .
For the system, to have a unique solution, we must have
$\frac{a_{1}}{a_{2}} \neq \frac{b_{1}}{b_{2}} \Rightarrow \frac{k}{6} \neq \frac{- 1}{- 2} = \frac{1}{2} \Rightarrow k \neq 3$
Hence, $k \neq 3$ .

Answer:

The given system is
$2 x + 3 y - 5 = 0 . . . . . (i) 4 x + k y - 10 = 0 . . . . . (i i)$
Here, $a_{1} = 2, b_{1} = 3, c_{1} = - 5 . a_{2} = 4, b_{2} = k and c_{2} = - 10$ .
For the system, to have an infinite number of solutions, we must have
$\frac{a_{1}}{a_{2}} = \frac{b_{1}}{b_{2}} = \frac{c_{1}}{c_{2}} \Rightarrow \frac{2}{4} = \frac{3}{k} = \frac{- 5}{- 10} \Rightarrow \frac{1}{2} = \frac{3}{k} = \frac{1}{2} \Rightarrow k = 6$
Hence, k = 6.

Question 19:

The given system is
$2 x + 3 y - 5 = 0 . . . . . (i) 4 x + k y - 10 = 0 . . . . . (i i)$
Here, $a_{1} = 2, b_{1} = 3, c_{1} = - 5 . a_{2} = 4, b_{2} = k and c_{2} = - 10$ .
For the system, to have an infinite number of solutions, we must have
$\frac{a_{1}}{a_{2}} = \frac{b_{1}}{b_{2}} = \frac{c_{1}}{c_{2}} \Rightarrow \frac{2}{4} = \frac{3}{k} = \frac{- 5}{- 10} \Rightarrow \frac{1}{2} = \frac{3}{k} = \frac{1}{2} \Rightarrow k = 6$
Hence, k = 6.

Answer:

The given system is
$2 x + 3 y - 1 = 0 . . . . . (i) 4 x + 6 y - 4 = 0 . . . . . (i i)$
Here, $a_{1} = 2, b_{1} = 3, c_{1} = - 1 . a_{2} = 4, b_{2} = 6 and c_{2} = - 4$ .
Now,
$\frac{a_{1}}{a_{2}} = \frac{2}{4} = \frac{1}{2} \frac{b_{1}}{b_{2}} = \frac{3}{6} = \frac{1}{2} \frac{c_{1}}{c_{2}} = \frac{- 1}{- 4} = \frac{1}{4}$
Thus, $\frac{a_{1}}{a_{2}} = \frac{b_{1}}{b_{2}} \neq \frac{c_{1}}{c_{2}}$ and therefor the given system has no solution.

Question 20:

The given system is
$2 x + 3 y - 1 = 0 . . . . . (i) 4 x + 6 y - 4 = 0 . . . . . (i i)$
Here, $a_{1} = 2, b_{1} = 3, c_{1} = - 1 . a_{2} = 4, b_{2} = 6 and c_{2} = - 4$ .
Now,
$\frac{a_{1}}{a_{2}} = \frac{2}{4} = \frac{1}{2} \frac{b_{1}}{b_{2}} = \frac{3}{6} = \frac{1}{2} \frac{c_{1}}{c_{2}} = \frac{- 1}{- 4} = \frac{1}{4}$
Thus, $\frac{a_{1}}{a_{2}} = \frac{b_{1}}{b_{2}} \neq \frac{c_{1}}{c_{2}}$ and therefor the given system has no solution.

Answer:

The given system is
$x + 2 y - 3 = 0 . . . . . (i) 5 x + k y + 7 = 0 . . . . . (i i)$
Here, $a_{1} = 1, b_{1} = 2, c_{1} = - 3, a_{2} = 5, b_{2} = k and c_{2} = 7$ .
For the system to be inconsistent, we must have
$\frac{a_{1}}{a_{2}} = \frac{b_{1}}{b_{2}} \neq \frac{c_{1}}{c_{2}} \Rightarrow \frac{1}{5} = \frac{2}{k} \neq \frac{- 3}{7} \Rightarrow \frac{1}{5} = \frac{2}{k} \Rightarrow k = 10$
Hence, k = 10.

Question 21:

The given system is
$x + 2 y - 3 = 0 . . . . . (i) 5 x + k y + 7 = 0 . . . . . (i i)$
Here, $a_{1} = 1, b_{1} = 2, c_{1} = - 3, a_{2} = 5, b_{2} = k and c_{2} = 7$ .
For the system to be inconsistent, we must have
$\frac{a_{1}}{a_{2}} = \frac{b_{1}}{b_{2}} \neq \frac{c_{1}}{c_{2}} \Rightarrow \frac{1}{5} = \frac{2}{k} \neq \frac{- 3}{7} \Rightarrow \frac{1}{5} = \frac{2}{k} \Rightarrow k = 10$
Hence, k = 10.

Answer:

The given system of equations is
$\frac{3}{x + y} + \frac{2}{x - y} = 2 . . . . . (i) \frac{9}{x + y} - \frac{4}{x - y} = 1 . . . . . (i i)$
Substituting $\frac{1}{x + y} = u and \frac{1}{x - y} = v$ in (i) and (ii), the given equations are changed to
$3 u + 2 v = 2 . . . . . (i i i) 9 u - 4 v = 1 . . . . . (i v)$
Multiplying (i) by 2 and adding it with (ii), we get
$15 u = 4 + 1 \Rightarrow u = \frac{1}{3}$
Multiplying (i) by 3 and subtracting (ii) from it, we get
$6 v + 4 v = 6 - 1 \Rightarrow u = \frac{5}{10} = \frac{1}{2}$
Therefore
$x + y = 3 . . . . . (v) x - y = 2 . . . . . (v i)$
Now, adding (v) and (vi) we have
$2 x = 5 \Rightarrow x = \frac{5}{2}$
Substituting $x = \frac{5}{2}$ in (v), we have
$\frac{5}{2} + y = 3 \Rightarrow y = 3 - \frac{5}{2} = \frac{1}{2}$
Hence, $x = \frac{5}{2} and y = \frac{1}{2}$ .

Question 19:

The given system of equations is
$\frac{3}{x + y} + \frac{2}{x - y} = 2 . . . . . (i) \frac{9}{x + y} - \frac{4}{x - y} = 1 . . . . . (i i)$
Substituting $\frac{1}{x + y} = u and \frac{1}{x - y} = v$ in (i) and (ii), the given equations are changed to
$3 u + 2 v = 2 . . . . . (i i i) 9 u - 4 v = 1 . . . . . (i v)$
Multiplying (i) by 2 and adding it with (ii), we get
$15 u = 4 + 1 \Rightarrow u = \frac{1}{3}$
Multiplying (i) by 3 and subtracting (ii) from it, we get
$6 v + 4 v = 6 - 1 \Rightarrow u = \frac{5}{10} = \frac{1}{2}$
Therefore
$x + y = 3 . . . . . (v) x - y = 2 . . . . . (v i)$
Now, adding (v) and (vi) we have
$2 x = 5 \Rightarrow x = \frac{5}{2}$
Substituting $x = \frac{5}{2}$ in (v), we have
$\frac{5}{2} + y = 3 \Rightarrow y = 3 - \frac{5}{2} = \frac{1}{2}$
Hence, $x = \frac{5}{2} and y = \frac{1}{2}$ .

Answer:

Let the required fraction be $\frac{x}{y}$ .
Then, we have:
$\frac{x + 1}{y + 1} = \frac{4}{5}$
⇒ 5(x + 1) = 4(y + 1)
⇒ 5x + 5 = 4y + 4
⇒ 5x − 4y = −1                       ....(i)

Again, we have:
$\frac{x - 5}{y - 5} = \frac{1}{2}$
⇒ 2(x − 5) = 1(y − 5)
⇒ 2x − 10 = y − 5
⇒ 2x − y = 5                         ....(ii)
On multiplying (ii) by 4, we get:
8x − 4y = 20                             ....(iii)
On subtracting (i) from (iii), we get:
3x = (20 − (−1)) = 20 + 1 = 21
⇒ 3x = 21
⇒ x = 7
On substituting x = 7 in (i), we get:
5 × 7 − 4y = −1
⇒ 35 − 4y = −1
⇒ 4y = 36
⇒ y = 9
∴ â€‹x = 7 and y = 9
Hence, the required fraction is $\frac{7}{9}$ .

Question 20:

Let the required fraction be $\frac{x}{y}$ .
Then, we have:
$\frac{x + 1}{y + 1} = \frac{4}{5}$
⇒ 5(x + 1) = 4(y + 1)
⇒ 5x + 5 = 4y + 4
⇒ 5x − 4y = −1                       ....(i)

Again, we have:
$\frac{x - 5}{y - 5} = \frac{1}{2}$
⇒ 2(x − 5) = 1(y − 5)
⇒ 2x − 10 = y − 5
⇒ 2x − y = 5                         ....(ii)
On multiplying (ii) by 4, we get:
8x − 4y = 20                             ....(iii)
On subtracting (i) from (iii), we get:
3x = (20 − (−1)) = 20 + 1 = 21
⇒ 3x = 21
⇒ x = 7
On substituting x = 7 in (i), we get:
5 × 7 − 4y = −1
⇒ 35 − 4y = −1
⇒ 4y = 36
⇒ y = 9
∴ â€‹x = 7 and y = 9
Hence, the required fraction is $\frac{7}{9}$ .

Answer:

The given equations may be written as follows:
$\frac{a x}{b} - \frac{b y}{a} - (a + b) = 0$ ....(i)
$a x - b y - 2 a b = 0$ ....(ii)
Here, a₁= $\frac{a}{b}$ , b₁ = $\frac{- b}{a}$ , c₁ = −(a + b), a₂ = a, b₂ = −b, c₂ = −2ab
By cross multiplication, we have:

∴ $\frac{x}{(- \frac{b}{a}) \times (- 2 a b) - (- b) \times (- (a + b))} = \frac{y}{- (a + b) \times a - (- 2 a b) \times \frac{a}{b}} = \frac{1}{\frac{a}{b} \times (- b) - a \times (- \frac{b}{a})}$
⇒ $\frac{x}{2 b^{2} - b (a + b)} = \frac{y}{- a (a + b) + 2 a^{2}} = \frac{1}{- a + b}$
⇒ $\frac{x}{2 b^{2} - a b - b^{2}} = \frac{y}{- a^{2} - a b + 2 a^{2}} = \frac{1}{- a + b}$
⇒ $\frac{x}{b^{2} - a b} = \frac{y}{a^{2} - a b} = \frac{1}{- (a - b)}$
⇒ $\frac{x}{- b (a - b)} = \frac{y}{a (a - b)} = \frac{1}{- (a - b)}$
⇒ $x = \frac{- b (a - b)}{- (a - b)} = b, y = \frac{a (a - b)}{- (a - b)} = - a$
Hence, x = b and y = −a is the required solution.

Question 1:

The given equations may be written as follows:
$\frac{a x}{b} - \frac{b y}{a} - (a + b) = 0$ ....(i)
$a x - b y - 2 a b = 0$ ....(ii)
Here, a₁= $\frac{a}{b}$ , b₁ = $\frac{- b}{a}$ , c₁ = −(a + b), a₂ = a, b₂ = −b, c₂ = −2ab
By cross multiplication, we have:

∴ $\frac{x}{(- \frac{b}{a}) \times (- 2 a b) - (- b) \times (- (a + b))} = \frac{y}{- (a + b) \times a - (- 2 a b) \times \frac{a}{b}} = \frac{1}{\frac{a}{b} \times (- b) - a \times (- \frac{b}{a})}$
⇒ $\frac{x}{2 b^{2} - b (a + b)} = \frac{y}{- a (a + b) + 2 a^{2}} = \frac{1}{- a + b}$
⇒ $\frac{x}{2 b^{2} - a b - b^{2}} = \frac{y}{- a^{2} - a b + 2 a^{2}} = \frac{1}{- a + b}$
⇒ $\frac{x}{b^{2} - a b} = \frac{y}{a^{2} - a b} = \frac{1}{- (a - b)}$
⇒ $\frac{x}{- b (a - b)} = \frac{y}{a (a - b)} = \frac{1}{- (a - b)}$
⇒ $x = \frac{- b (a - b)}{- (a - b)} = b, y = \frac{a (a - b)}{- (a - b)} = - a$
Hence, x = b and y = −a is the required solution.

Answer:

The given system of equations is
$2 x + 3 y = 12 . . . . . (i) 3 x - 2 y = 5 . . . . . (i i)$
Multiplying (i) by 2 and (ii) by 3 and then adding, we get
$4 x + 9 x = 24 + 15 \Rightarrow x = \frac{39}{13} = 3$
Now, putting x = 3 in (i), we have
$2 \times 3 + 3 y = 12 \Rightarrow y = \frac{12 - 6}{3} = 2$
Thus, x = 3 and y = 2.
Hence, the correct answer is option (c).

Question 2:

The given system of equations is
$2 x + 3 y = 12 . . . . . (i) 3 x - 2 y = 5 . . . . . (i i)$
Multiplying (i) by 2 and (ii) by 3 and then adding, we get
$4 x + 9 x = 24 + 15 \Rightarrow x = \frac{39}{13} = 3$
Now, putting x = 3 in (i), we have
$2 \times 3 + 3 y = 12 \Rightarrow y = \frac{12 - 6}{3} = 2$
Thus, x = 3 and y = 2.
Hence, the correct answer is option (c).

Answer:

The given system of equations is
$x - y = 2 . . . . . (i) x + y = 10 . . . . . (i i)$
Adding (i) and (ii), we get
$2 x = 12 \Rightarrow x = 6$
Now, putting x = 6 in (ii), we have
$6 + y = 10 \Rightarrow y = 10 - 6 = 4$
Thus, x = 6 and y = 4.
Hence, the correct answer is option (c).

Question 3:

The given system of equations is
$x - y = 2 . . . . . (i) x + y = 10 . . . . . (i i)$
Adding (i) and (ii), we get
$2 x = 12 \Rightarrow x = 6$
Now, putting x = 6 in (ii), we have
$6 + y = 10 \Rightarrow y = 10 - 6 = 4$
Thus, x = 6 and y = 4.
Hence, the correct answer is option (c).

Answer:

The given system of equations is
$\frac{2 x}{3} - \frac{y}{2} = - \frac{1}{6} . . . . . (i) \frac{x}{2} + \frac{2 y}{3} = 3 . . . . . (i i)$
Multiplying (i) and (ii) by 6, we get
$4 x - 3 y = - 1 . . . . . (i i i) 3 x + 4 y = 18 . . . . . (i v)$
Multiplying (iii) by 4 and (iv) by 3 and adding, we get
$16 x + 9 x = - 4 + 54 \Rightarrow x = \frac{50}{25} = 2$
Now, putting x = 2 in (iv), we have
$3 \times 2 + 4 y = 18 \Rightarrow y = \frac{18 - 6}{4} = 3$
Thus, x = 2 and y = 3.
Hence, the correct answer is option (a).

Question 4:

The given system of equations is
$\frac{2 x}{3} - \frac{y}{2} = - \frac{1}{6} . . . . . (i) \frac{x}{2} + \frac{2 y}{3} = 3 . . . . . (i i)$
Multiplying (i) and (ii) by 6, we get
$4 x - 3 y = - 1 . . . . . (i i i) 3 x + 4 y = 18 . . . . . (i v)$
Multiplying (iii) by 4 and (iv) by 3 and adding, we get
$16 x + 9 x = - 4 + 54 \Rightarrow x = \frac{50}{25} = 2$
Now, putting x = 2 in (iv), we have
$3 \times 2 + 4 y = 18 \Rightarrow y = \frac{18 - 6}{4} = 3$
Thus, x = 2 and y = 3.
Hence, the correct answer is option (a).

Answer:

The given system of equations is
$\frac{1}{x} + \frac{2}{y} = 4 . . . . . (i) \frac{3}{y} - \frac{1}{x} = 11 . . . . . (i i)$
Adding (i) and (ii), we get
$\frac{2}{y} + \frac{3}{y} = 15 \Rightarrow \frac{5}{y} = 15 \Rightarrow y = \frac{5}{15} = \frac{1}{3}$
Now, putting $y = \frac{1}{3}$ in (i), we have
$\frac{1}{x} + 2 \times 3 = 4 \Rightarrow \frac{1}{x} = 4 - 6 \Rightarrow x = - \frac{1}{2}$
Thus, $x = - \frac{1}{2}, y = \frac{1}{3}$ .
Hence, the correct answer is option (d).

Question 5:

The given system of equations is
$\frac{1}{x} + \frac{2}{y} = 4 . . . . . (i) \frac{3}{y} - \frac{1}{x} = 11 . . . . . (i i)$
Adding (i) and (ii), we get
$\frac{2}{y} + \frac{3}{y} = 15 \Rightarrow \frac{5}{y} = 15 \Rightarrow y = \frac{5}{15} = \frac{1}{3}$
Now, putting $y = \frac{1}{3}$ in (i), we have
$\frac{1}{x} + 2 \times 3 = 4 \Rightarrow \frac{1}{x} = 4 - 6 \Rightarrow x = - \frac{1}{2}$
Thus, $x = - \frac{1}{2}, y = \frac{1}{3}$ .
Hence, the correct answer is option (d).

Answer:

Consider $\frac{2 x + y + 2}{5} = \frac{3 x - y + 1}{3} and \frac{3 x - y + 1}{3} = \frac{3 x + 2 y + 1}{6}$ . Now, simplifying these equations, we get
$3 (2 x + y + 2) = 5 (3 x - y + 1) \Rightarrow 6 x + 3 y + 6 = 15 x - 5 y + 5 \Rightarrow 9 x - 8 y = 1 . . . . . (i)$
And
$6 (3 x - y + 1) = 3 (3 x + 2 y + 1) \Rightarrow 18 x - 6 y + 6 = 9 x + 6 y + 3 \Rightarrow 3 x - 4 y = - 1 . . . . . (i i)$
Multiplying (ii) by 2 and subtracting it from (i)
$9 x - 6 x = 1 + 2 \Rightarrow x = 1$
Now, putting x = 1 in (ii), we have
$3 \times 1 - 4 y = - 1 \Rightarrow y = \frac{3 + 1}{4} = 1$
Thus, x = 1, y = 1.
Hence, the correct answer is option (a).

Question 6:

Consider $\frac{2 x + y + 2}{5} = \frac{3 x - y + 1}{3} and \frac{3 x - y + 1}{3} = \frac{3 x + 2 y + 1}{6}$ . Now, simplifying these equations, we get
$3 (2 x + y + 2) = 5 (3 x - y + 1) \Rightarrow 6 x + 3 y + 6 = 15 x - 5 y + 5 \Rightarrow 9 x - 8 y = 1 . . . . . (i)$
And
$6 (3 x - y + 1) = 3 (3 x + 2 y + 1) \Rightarrow 18 x - 6 y + 6 = 9 x + 6 y + 3 \Rightarrow 3 x - 4 y = - 1 . . . . . (i i)$
Multiplying (ii) by 2 and subtracting it from (i)
$9 x - 6 x = 1 + 2 \Rightarrow x = 1$
Now, putting x = 1 in (ii), we have
$3 \times 1 - 4 y = - 1 \Rightarrow y = \frac{3 + 1}{4} = 1$
Thus, x = 1, y = 1.
Hence, the correct answer is option (a).

Answer:

The given equations are
$\frac{3}{x + y} + \frac{2}{x - y} = 2 . . . . . (i) \frac{9}{x + y} - \frac{4}{x - y} = 1 . . . . . (i i)$
Substituting $\frac{1}{x + y} = u and \frac{1}{x - y} = v$ in (i) and (ii), the new system becomes
$3 u + 2 v = 2 . . . . . (i i i) 9 u - 4 v = 1 . . . . . (i v)$
Now, multiplying (iii) by 2 and adding it with (iv), we get
$6 u + 9 u = 4 + 1 \Rightarrow u = \frac{5}{15} = \frac{1}{3}$
Again, multiplying (iii) by 3 and subtracting (iv) from it, we get
$6 v + 4 v = 6 - 1 \Rightarrow v = \frac{5}{10} = \frac{1}{2}$
Therefore
$x + y = 3 . . . . . (v) x - y = 2 . . . . . (v i)$
Adding (v) and (vi), we get
$2 x = 3 + 2 \Rightarrow x = \frac{5}{2}$
Substituting $x = \frac{5}{2}$ , in (v), we have
$\frac{5}{2} + y = 3 \Rightarrow y = 3 - \frac{5}{2} = \frac{1}{2}$
Thus, $x = \frac{5}{2} and y = \frac{1}{2}$ .
Hence, the correct answer is option (b).

Question 7:

The given equations are
$\frac{3}{x + y} + \frac{2}{x - y} = 2 . . . . . (i) \frac{9}{x + y} - \frac{4}{x - y} = 1 . . . . . (i i)$
Substituting $\frac{1}{x + y} = u and \frac{1}{x - y} = v$ in (i) and (ii), the new system becomes
$3 u + 2 v = 2 . . . . . (i i i) 9 u - 4 v = 1 . . . . . (i v)$
Now, multiplying (iii) by 2 and adding it with (iv), we get
$6 u + 9 u = 4 + 1 \Rightarrow u = \frac{5}{15} = \frac{1}{3}$
Again, multiplying (iii) by 3 and subtracting (iv) from it, we get
$6 v + 4 v = 6 - 1 \Rightarrow v = \frac{5}{10} = \frac{1}{2}$
Therefore
$x + y = 3 . . . . . (v) x - y = 2 . . . . . (v i)$
Adding (v) and (vi), we get
$2 x = 3 + 2 \Rightarrow x = \frac{5}{2}$
Substituting $x = \frac{5}{2}$ , in (v), we have
$\frac{5}{2} + y = 3 \Rightarrow y = 3 - \frac{5}{2} = \frac{1}{2}$
Thus, $x = \frac{5}{2} and y = \frac{1}{2}$ .
Hence, the correct answer is option (b).

Answer:

The given equations are
$4 x + 6 y = 3 x y . . . . . (i) 8 x + 9 y = 5 x y . . . . . (i i)$
Dividing (i) and (ii) by xy, we get
$\frac{6}{x} + \frac{4}{y} = 3 . . . . . (i i i) \frac{9}{x} + \frac{8}{y} = 5 . . . . . (i v)$
Multiplying (iii) by 2 and subtracting (iv) from it, we get
$\frac{12}{x} - \frac{9}{x} = 6 - 5 \Rightarrow \frac{3}{x} = 1 \Rightarrow x = 3$
Substituting x = 3 in (iii), we get
$\frac{6}{3} + \frac{4}{y} = 3 \Rightarrow \frac{4}{y} = 1 \Rightarrow y = 4$
Thus, x = 3 and y = 4.
Hence, the correct answer is option (c).

Question 8:

The given equations are
$4 x + 6 y = 3 x y . . . . . (i) 8 x + 9 y = 5 x y . . . . . (i i)$
Dividing (i) and (ii) by xy, we get
$\frac{6}{x} + \frac{4}{y} = 3 . . . . . (i i i) \frac{9}{x} + \frac{8}{y} = 5 . . . . . (i v)$
Multiplying (iii) by 2 and subtracting (iv) from it, we get
$\frac{12}{x} - \frac{9}{x} = 6 - 5 \Rightarrow \frac{3}{x} = 1 \Rightarrow x = 3$
Substituting x = 3 in (iii), we get
$\frac{6}{3} + \frac{4}{y} = 3 \Rightarrow \frac{4}{y} = 1 \Rightarrow y = 4$
Thus, x = 3 and y = 4.
Hence, the correct answer is option (c).

Answer:

The given equations are
$29 x + 37 y = 103 . . . . . (i) 37 x + 29 y = 95 . . . . . (i i)$
Adding (i) and (ii), we get
$66 x + 66 y = 198 \Rightarrow x + y = 3 . . . . . (i i i)$
Subtracting (i) from (ii), we get
$8 x - 8 y = - 8 \Rightarrow x - y = - 1 . . . . . (i v)$
Adding (iii) and (iv), we get
$2 x = 2 \Rightarrow x = 1$
Substituting x = 1 in (iii), we have
$1 + y = 3 \Rightarrow y = 2$
Thus, x = 1 and y = 2.
Hence, the correct answer is option (a).

Question 9:

The given equations are
$29 x + 37 y = 103 . . . . . (i) 37 x + 29 y = 95 . . . . . (i i)$
Adding (i) and (ii), we get
$66 x + 66 y = 198 \Rightarrow x + y = 3 . . . . . (i i i)$
Subtracting (i) from (ii), we get
$8 x - 8 y = - 8 \Rightarrow x - y = - 1 . . . . . (i v)$
Adding (iii) and (iv), we get
$2 x = 2 \Rightarrow x = 1$
Substituting x = 1 in (iii), we have
$1 + y = 3 \Rightarrow y = 2$
Thus, x = 1 and y = 2.
Hence, the correct answer is option (a).

Answer:

$∵ 2^{x + y} = 2^{x - y} = \sqrt{8} ∴ x + y = x - y \Rightarrow y = 0$
Hence, the correct answer is option (c).

Question 10:

$∵ 2^{x + y} = 2^{x - y} = \sqrt{8} ∴ x + y = x - y \Rightarrow y = 0$
Hence, the correct answer is option (c).

Answer:

The given equations are
$\frac{2}{x} + \frac{3}{y} = 6 . . . . . (i) \frac{1}{x} + \frac{1}{2 y} = 2 . . . . . (i i)$
Multiplying (ii) by 2 and subtracting it from (ii), we get
$\frac{3}{y} - \frac{1}{y} = 6 - 4 \Rightarrow \frac{2}{y} = 2 \Rightarrow y = 1$
Substituting y = 1 in (ii), we get
$\frac{1}{x} + \frac{1}{2} = 2 \Rightarrow \frac{1}{x} = 2 - \frac{1}{2} = \frac{3}{2} \Rightarrow x = \frac{2}{3}$
Hence, the correct answer is option (b).

Question 11:

The given equations are
$\frac{2}{x} + \frac{3}{y} = 6 . . . . . (i) \frac{1}{x} + \frac{1}{2 y} = 2 . . . . . (i i)$
Multiplying (ii) by 2 and subtracting it from (ii), we get
$\frac{3}{y} - \frac{1}{y} = 6 - 4 \Rightarrow \frac{2}{y} = 2 \Rightarrow y = 1$
Substituting y = 1 in (ii), we get
$\frac{1}{x} + \frac{1}{2} = 2 \Rightarrow \frac{1}{x} = 2 - \frac{1}{2} = \frac{3}{2} \Rightarrow x = \frac{2}{3}$
Hence, the correct answer is option (b).

Answer:

The given equations are
$k x - y - 2 = 0 . . . . . (i) 6 x - 2 y - 3 = 0 . . . . . (i i)$
Here, $a_{1} = k, b_{1} = - 1, c_{1} = - 2, a_{2} = 6, b_{2} = - 2 and c_{2} = - 3 .$
For the given system to have a unique solution, we must have
$\frac{a_{1}}{a_{2}} \neq \frac{b_{1}}{b_{2}} \Rightarrow \frac{k}{6} \neq \frac{- 1}{- 2} \Rightarrow k \neq 3$
Hence, the correct answer is option (d).

Question 12:

The given equations are
$k x - y - 2 = 0 . . . . . (i) 6 x - 2 y - 3 = 0 . . . . . (i i)$
Here, $a_{1} = k, b_{1} = - 1, c_{1} = - 2, a_{2} = 6, b_{2} = - 2 and c_{2} = - 3 .$
For the given system to have a unique solution, we must have
$\frac{a_{1}}{a_{2}} \neq \frac{b_{1}}{b_{2}} \Rightarrow \frac{k}{6} \neq \frac{- 1}{- 2} \Rightarrow k \neq 3$
Hence, the correct answer is option (d).

Answer:

The correct option is (b).

The given system of equations can be written as follows:
x − 2y − 3 = 0 and 3x + ky − 1 = 0
The given equations are of the following form:
a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0
Here, a₁ = 1, b₁ = −2, c₁ = −3 and a₂ = 3, b₂ = k and c₂ = −1
∴ $\frac{a_{1}}{a_{2}} = \frac{1}{3}, \frac{b_{1}}{b_{2}} = \frac{- 2}{k} and \frac{c_{1}}{c_{2}} = \frac{- 3}{- 1} = 3$
These graph lines will intersect at a unique point when we have:
$\frac{a_{1}}{a_{2}} \neq \frac{b_{1}}{b_{2}}$ ⇒ $\frac{1}{3} \neq \frac{- 2}{k} \Rightarrow k \neq - 6$
Hence, k has all real values other than −6.

Question 13:

The correct option is (b).

The given system of equations can be written as follows:
x − 2y − 3 = 0 and 3x + ky − 1 = 0
The given equations are of the following form:
a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0
Here, a₁ = 1, b₁ = −2, c₁ = −3 and a₂ = 3, b₂ = k and c₂ = −1
∴ $\frac{a_{1}}{a_{2}} = \frac{1}{3}, \frac{b_{1}}{b_{2}} = \frac{- 2}{k} and \frac{c_{1}}{c_{2}} = \frac{- 3}{- 1} = 3$
These graph lines will intersect at a unique point when we have:
$\frac{a_{1}}{a_{2}} \neq \frac{b_{1}}{b_{2}}$ ⇒ $\frac{1}{3} \neq \frac{- 2}{k} \Rightarrow k \neq - 6$
Hence, k has all real values other than −6.

Answer:

The correct option is (a).

The given system of equations can be written as follows:
x + 2y − 3 = 0 and 5x + ky + 7 = 0
The given equations are of the following form:
a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0
Here, a₁ = 1, b₁ = 2, c₁ = −3 and a₂ = 5, b₂ = k and c₂ = 7
∴ $\frac{a_{1}}{a_{2}} = \frac{1}{5}, \frac{b_{1}}{b_{2}} = \frac{2}{k} and \frac{c_{1}}{c_{2}} = \frac{- 3}{7}$
For the system of equations to have no solution, we must have:
$\frac{a_{1}}{a_{2}} = \frac{b_{1}}{b_{2}} \neq \frac{c_{1}}{c_{2}}$
∴ $\frac{1}{5} = \frac{2}{k} \neq \frac{- 3}{7} \Rightarrow k = 10$

Question 14:

The correct option is (a).

The given system of equations can be written as follows:
x + 2y − 3 = 0 and 5x + ky + 7 = 0
The given equations are of the following form:
a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0
Here, a₁ = 1, b₁ = 2, c₁ = −3 and a₂ = 5, b₂ = k and c₂ = 7
∴ $\frac{a_{1}}{a_{2}} = \frac{1}{5}, \frac{b_{1}}{b_{2}} = \frac{2}{k} and \frac{c_{1}}{c_{2}} = \frac{- 3}{7}$
For the system of equations to have no solution, we must have:
$\frac{a_{1}}{a_{2}} = \frac{b_{1}}{b_{2}} \neq \frac{c_{1}}{c_{2}}$
∴ $\frac{1}{5} = \frac{2}{k} \neq \frac{- 3}{7} \Rightarrow k = 10$

Answer:

The correct option is (d).

The given system of equations can be written as follows:
3x + 2ky − 2 = 0 and 2x + 5y + 1 = 0
The given equations are of the following form:
a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0
Here, a₁ = 3, b₁ = 2k, c₁ = −2 and a₂ = 2, b₂ = 5 and c₂ = 1
∴ $\frac{a_{1}}{a_{2}} = \frac{3}{2}, \frac{b_{1}}{b_{2}} = \frac{2 k}{5} and \frac{c_{1}}{c_{2}} = \frac{- 2}{1}$
For parallel lines, we have:
$\frac{a_{1}}{a_{2}} = \frac{b_{1}}{b_{2}} \neq \frac{c_{1}}{c_{2}}$
∴ $\frac{3}{2} = \frac{2 k}{5} \neq \frac{- 2}{1}$
⇒ $k = \frac{15}{4}$

Question 15:

The correct option is (d).

The given system of equations can be written as follows:
3x + 2ky − 2 = 0 and 2x + 5y + 1 = 0
The given equations are of the following form:
a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0
Here, a₁ = 3, b₁ = 2k, c₁ = −2 and a₂ = 2, b₂ = 5 and c₂ = 1
∴ $\frac{a_{1}}{a_{2}} = \frac{3}{2}, \frac{b_{1}}{b_{2}} = \frac{2 k}{5} and \frac{c_{1}}{c_{2}} = \frac{- 2}{1}$
For parallel lines, we have:
$\frac{a_{1}}{a_{2}} = \frac{b_{1}}{b_{2}} \neq \frac{c_{1}}{c_{2}}$
∴ $\frac{3}{2} = \frac{2 k}{5} \neq \frac{- 2}{1}$
⇒ $k = \frac{15}{4}$

Answer:

The correct option is (d).

The given system of equations can be written as follows:
kx − 2y − 3 = 0 and 3x + y − 5 = 0
The given equations are of the following form:
a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0
Here, a₁ = k, b₁ = −2, c₁ = −3 and a₂ = 3, b₂ = 1 and c₂ = −5
∴ $\frac{a_{1}}{a_{2}} = \frac{k}{3}, \frac{b_{1}}{b_{2}} = \frac{- 2}{1} and \frac{c_{1}}{c_{2}} = \frac{- 3}{- 5} = \frac{3}{5}$
Thus, for these graph lines to intersect at a unique point, we must have:
$\frac{a_{1}}{a_{2}} \neq \frac{b_{1}}{b_{2}}$
⇒ $\frac{k}{3} \neq \frac{- 2}{1} \Rightarrow k \neq - 6$
Hence, the graph lines will intersect at all real values of k except −6.

Question 16:

The correct option is (d).

The given system of equations can be written as follows:
kx − 2y − 3 = 0 and 3x + y − 5 = 0
The given equations are of the following form:
a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0
Here, a₁ = k, b₁ = −2, c₁ = −3 and a₂ = 3, b₂ = 1 and c₂ = −5
∴ $\frac{a_{1}}{a_{2}} = \frac{k}{3}, \frac{b_{1}}{b_{2}} = \frac{- 2}{1} and \frac{c_{1}}{c_{2}} = \frac{- 3}{- 5} = \frac{3}{5}$
Thus, for these graph lines to intersect at a unique point, we must have:
$\frac{a_{1}}{a_{2}} \neq \frac{b_{1}}{b_{2}}$
⇒ $\frac{k}{3} \neq \frac{- 2}{1} \Rightarrow k \neq - 6$
Hence, the graph lines will intersect at all real values of k except −6.

Answer:

The correct option is (d).

The given system of equations can be written as:
x + 2y + 5 = 0 and −3x − 6y + 1 = 0
The given equations are of the following form:
a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0
Here, a₁ = 1, b₁ = 2, c₁ = 5 and a₂ = −3, b₂ = −6 and c₂ = 1
∴ $\frac{a_{1}}{a_{2}} = \frac{1}{- 3}, \frac{b_{1}}{b_{2}} = \frac{2}{- 6} = \frac{1}{- 3} and \frac{c_{1}}{c_{2}} = \frac{5}{1}$
∴ $\frac{a_{1}}{a_{2}} = \frac{b_{1}}{b_{2}} \neq \frac{c_{1}}{c_{2}}$
Hence, the given system has no solution.

Question 17:

The correct option is (d).

The given system of equations can be written as:
x + 2y + 5 = 0 and −3x − 6y + 1 = 0
The given equations are of the following form:
a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0
Here, a₁ = 1, b₁ = 2, c₁ = 5 and a₂ = −3, b₂ = −6 and c₂ = 1
∴ $\frac{a_{1}}{a_{2}} = \frac{1}{- 3}, \frac{b_{1}}{b_{2}} = \frac{2}{- 6} = \frac{1}{- 3} and \frac{c_{1}}{c_{2}} = \frac{5}{1}$
∴ $\frac{a_{1}}{a_{2}} = \frac{b_{1}}{b_{2}} \neq \frac{c_{1}}{c_{2}}$
Hence, the given system has no solution.

Answer:

The correct option is (d).

The given system of equations can be written as follows:
2x + 3y − 5 = 0 and 4x + 6y − 15 = 0
The given equations are of the following form:
a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0
Here, a₁ = 2, b₁ = 3, c₁ = −5 and a₂ = 4, b₂ = 6 and c₂ = −15
∴ $\frac{a_{1}}{a_{2}} = \frac{2}{4} = \frac{1}{2}, \frac{b_{1}}{b_{2}} = \frac{3}{6} = \frac{1}{2} and \frac{c_{1}}{c_{2}} = \frac{- 5}{- 15} = \frac{1}{3}$
∴ â€‹ $\frac{a_{1}}{a_{2}} = \frac{b_{1}}{b_{2}} \neq \frac{c_{1}}{c_{2}}$
Hence, the given system has no solution.

Question 18:

The correct option is (d).

The given system of equations can be written as follows:
2x + 3y − 5 = 0 and 4x + 6y − 15 = 0
The given equations are of the following form:
a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0
Here, a₁ = 2, b₁ = 3, c₁ = −5 and a₂ = 4, b₂ = 6 and c₂ = −15
∴ $\frac{a_{1}}{a_{2}} = \frac{2}{4} = \frac{1}{2}, \frac{b_{1}}{b_{2}} = \frac{3}{6} = \frac{1}{2} and \frac{c_{1}}{c_{2}} = \frac{- 5}{- 15} = \frac{1}{3}$
∴ â€‹ $\frac{a_{1}}{a_{2}} = \frac{b_{1}}{b_{2}} \neq \frac{c_{1}}{c_{2}}$
Hence, the given system has no solution.

Answer:

The correct option is (d).

If a pair of linear equations is consistent, then the two graph lines either intersect at a point or coincide.

Question 19:

The correct option is (d).

If a pair of linear equations is consistent, then the two graph lines either intersect at a point or coincide.

Answer:

The correct option is (a).

If a pair of linear equations in two variables is inconsistent, then no solution exists as they have no common point. And, since there is no common solution, their graph lines do not intersect. Hence, they are parallel.

Question 20:

The correct option is (a).

If a pair of linear equations in two variables is inconsistent, then no solution exists as they have no common point. And, since there is no common solution, their graph lines do not intersect. Hence, they are parallel.

Answer:

The correct option is (b).

Let $∠ A = x ° a n d ∠ B = y °$
∴ $∠ C = 3 ∠ B = (3 y) °$
Now, $∠ A + ∠ B + ∠ C = 180 °$
⇒ x + y + 3y = 180
⇒ x + 4y = 180              ...(i)
Also, $∠ C = 2 (∠ A + ∠ B)$
⇒ 3y = 2(x + y)
⇒ 2x − y = 0                  ...(ii)
On multiplying (ii) by 4, we get:
8x − 4y = 0                    ...(iii)
On adding (i) and (iii) we get:
9x = 180 ⇒ x = 20
On substituting x = 20 in (i), we get:
20 + 4y = 180 ⇒ 4y = (180 − 20) = 160 ⇒ y = 40
∴ x = 20 and y = 40
∴â€‹ $∠ B = y ° = 40 °$

Question 21:

The correct option is (b).

Let $∠ A = x ° a n d ∠ B = y °$
∴ $∠ C = 3 ∠ B = (3 y) °$
Now, $∠ A + ∠ B + ∠ C = 180 °$
⇒ x + y + 3y = 180
⇒ x + 4y = 180              ...(i)
Also, $∠ C = 2 (∠ A + ∠ B)$
⇒ 3y = 2(x + y)
⇒ 2x − y = 0                  ...(ii)
On multiplying (ii) by 4, we get:
8x − 4y = 0                    ...(iii)
On adding (i) and (iii) we get:
9x = 180 ⇒ x = 20
On substituting x = 20 in (i), we get:
20 + 4y = 180 ⇒ 4y = (180 − 20) = 160 ⇒ y = 40
∴ x = 20 and y = 40
∴â€‹ $∠ B = y ° = 40 °$

Answer:

The correct option is (b).
In a cyclic quadrilateral ABCD:
$∠ A = (x + y + 10) °$
$∠ B = (y + 20 °)$
$∠ C = (x + y - 30) °$
$∠ D = (x + y) °$
We have:
$∠ A + ∠ C = 180 °$ and $∠ B + ∠ D = 180 °$       [Since ABCD is a cyclic quadrilateral]
Now, $∠ A + ∠ C = (x + y + 10) ° + (x + y - 30) ° = 180 °$
⇒ 2x + 2y − 20 = 180
⇒ x + y − 10 = 90
⇒ x + y = 100                   ....(i)
Also, $∠ B + ∠ D = (y + 20) ° + (x + y) ° = 180 °$
⇒ x + 2y + 20 = 180
⇒ x + 2y = 160                ....(ii)
On subtracting (i) from (ii), we get:
y = (160 − 100) = 60
On substituting y = 60 in (i), we get:
x + 60 = 100 ⇒ x = (100 − 60) = 40
∴â€‹ $∠ B = (y + 20 °) = (60 + 20) ° = 80 °$

Question 22:

The correct option is (b).
In a cyclic quadrilateral ABCD:
$∠ A = (x + y + 10) °$
$∠ B = (y + 20 °)$
$∠ C = (x + y - 30) °$
$∠ D = (x + y) °$
We have:
$∠ A + ∠ C = 180 °$ and $∠ B + ∠ D = 180 °$       [Since ABCD is a cyclic quadrilateral]
Now, $∠ A + ∠ C = (x + y + 10) ° + (x + y - 30) ° = 180 °$
⇒ 2x + 2y − 20 = 180
⇒ x + y − 10 = 90
⇒ x + y = 100                   ....(i)
Also, $∠ B + ∠ D = (y + 20) ° + (x + y) ° = 180 °$
⇒ x + 2y + 20 = 180
⇒ x + 2y = 160                ....(ii)
On subtracting (i) from (ii), we get:
y = (160 − 100) = 60
On substituting y = 60 in (i), we get:
x + 60 = 100 ⇒ x = (100 − 60) = 40
∴â€‹ $∠ B = (y + 20 °) = (60 + 20) ° = 80 °$

Answer:

The correct option is (d).

Let the tens and the units digits of the required number be x and yâ€‹, respectively.
Required number = (10x + y)
According to the question, we have:
x + y = 15 ....(i)
Number obtained on reversing its digits = (10y + x)
∴ (10y + x) = (10x + y) + 9
⇒ 10y + x − 10x − y = 9
⇒ 9y − 9x = 9
⇒ y − x = 1 ....(ii)
On adding (i) and (ii), we get:
2y = 16 ⇒ y = 8
On substituting y = 8 in (i), we get:
x + 8 = 15 ⇒ x = (15 − 8) = 7
Number = (10x + y) = 10 × 7 + 8 = 70 + 8 = 78
Hence, the required number is 78.

Question 23:

The correct option is (d).

Let the tens and the units digits of the required number be x and yâ€‹, respectively.
Required number = (10x + y)
According to the question, we have:
x + y = 15 ....(i)
Number obtained on reversing its digits = (10y + x)
∴ (10y + x) = (10x + y) + 9
⇒ 10y + x − 10x − y = 9
⇒ 9y − 9x = 9
⇒ y − x = 1 ....(ii)
On adding (i) and (ii), we get:
2y = 16 ⇒ y = 8
On substituting y = 8 in (i), we get:
x + 8 = 15 ⇒ x = (15 − 8) = 7
Number = (10x + y) = 10 × 7 + 8 = 70 + 8 = 78
Hence, the required number is 78.

Answer:

Let the fraction be $\frac{x}{y}$
It is given that $\frac{x - 1}{y + 2} = \frac{1}{2}$
$\Rightarrow 2 x - 2 = y + 2 \Rightarrow 2 x - y = 4 . . . . . (i)$
Also, $\frac{x - 7}{y - 2} = \frac{1}{3}$
$\Rightarrow 3 x - 21 = y - 2 \Rightarrow 3 x - y = 19 . . . . . (ii)$
Subtract (ii) from (i), we get
$x = 15$
Put the value of x in equation (i), we get;
$2 (15) - y = 4 30 - y = 4 \Rightarrow y = 30 - 4 = 26$
Therefore, the fraction is $\frac{x}{y} = \frac{15}{26}$
Hence, the correct answer is option (b)

Question 24:

Let the fraction be $\frac{x}{y}$
It is given that $\frac{x - 1}{y + 2} = \frac{1}{2}$
$\Rightarrow 2 x - 2 = y + 2 \Rightarrow 2 x - y = 4 . . . . . (i)$
Also, $\frac{x - 7}{y - 2} = \frac{1}{3}$
$\Rightarrow 3 x - 21 = y - 2 \Rightarrow 3 x - y = 19 . . . . . (ii)$
Subtract (ii) from (i), we get
$x = 15$
Put the value of x in equation (i), we get;
$2 (15) - y = 4 30 - y = 4 \Rightarrow y = 30 - 4 = 26$
Therefore, the fraction is $\frac{x}{y} = \frac{15}{26}$
Hence, the correct answer is option (b)

Answer:

The correct option is (d).
Let the man's present age be x years.
Let his son's present age be y years.
Five years later:
(x + 5) = 3(y + 5)
⇒ x + 5 = 3y + 15
⇒ x − 3y = 10 ....(i)
Five years ago:
(x − 5) = 7(y − 5)
⇒ x − 5 = 7y − 35
⇒ x − 7y = −30 ....(ii)
On subtracting (i) from (ii), we get:
−4y = −40 ⇒ y = 10
On substituting y = 10 in (i), we get:
x − 3 × 10 = 10 ⇒ x − 30 = 10 ⇒ x = (10 + 30) = 40 years
Hence, the man's present age is 40 years.

Question 25:

The correct option is (d).
Let the man's present age be x years.
Let his son's present age be y years.
Five years later:
(x + 5) = 3(y + 5)
⇒ x + 5 = 3y + 15
⇒ x − 3y = 10 ....(i)
Five years ago:
(x − 5) = 7(y − 5)
⇒ x − 5 = 7y − 35
⇒ x − 7y = −30 ....(ii)
On subtracting (i) from (ii), we get:
−4y = −40 ⇒ y = 10
On substituting y = 10 in (i), we get:
x − 3 × 10 = 10 ⇒ x − 30 = 10 ⇒ x = (10 + 30) = 40 years
Hence, the man's present age is 40 years.

Answer:

The correct option is (b).

The given equations are as follows:
$6 x - 2 y + 9 = 0 and 3 x - y + 12 = 0$
They are of the following form:
a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0
Here, a₁ = 6, b₁ = −2, c₁ = 9 and a₂ = 3, b₂ = −1 and c₂ = 12
∴ $\frac{a_{1}}{a_{2}} = \frac{6}{3} = \frac{2}{1}, \frac{b_{1}}{b_{2}} = \frac{- 2}{- 1} = \frac{2}{1} and \frac{c_{1}}{c_{2}} = \frac{9}{12} = \frac{3}{4}$
∴ $\frac{a_{1}}{a_{2}} = \frac{b_{1}}{b_{2}} \neq \frac{c_{1}}{c_{2}}$
The given system has no solution.
Hence, the lines are parallel.

Question 26:

The correct option is (b).

The given equations are as follows:
$6 x - 2 y + 9 = 0 and 3 x - y + 12 = 0$
They are of the following form:
a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0
Here, a₁ = 6, b₁ = −2, c₁ = 9 and a₂ = 3, b₂ = −1 and c₂ = 12
∴ $\frac{a_{1}}{a_{2}} = \frac{6}{3} = \frac{2}{1}, \frac{b_{1}}{b_{2}} = \frac{- 2}{- 1} = \frac{2}{1} and \frac{c_{1}}{c_{2}} = \frac{9}{12} = \frac{3}{4}$
∴ $\frac{a_{1}}{a_{2}} = \frac{b_{1}}{b_{2}} \neq \frac{c_{1}}{c_{2}}$
The given system has no solution.
Hence, the lines are parallel.

Answer:

The correct option is (c).

The given equations are as follows:
$2 x + 3 y - 2 = 0 and x - 2 y - 8 = 0$
They are of the following form:
a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0
Here, a₁ = 2, b₁ = 3, c₁ = −2 and a₂ = 1, b₂ = −2 and c₂ = −8
∴ $\frac{a_{1}}{a_{2}} = \frac{2}{1}, \frac{b_{1}}{b_{2}} = \frac{3}{- 2} and \frac{c_{1}}{c_{2}} = \frac{- 2}{- 8} = \frac{1}{4}$
∴ $\frac{a_{1}}{a_{2}} \neq \frac{b_{1}}{b_{2}}$
The given system has a unique solution.
Hence, the lines intersect exactly at one point.

Question 27:

The correct option is (c).

The given equations are as follows:
$2 x + 3 y - 2 = 0 and x - 2 y - 8 = 0$
They are of the following form:
a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0
Here, a₁ = 2, b₁ = 3, c₁ = −2 and a₂ = 1, b₂ = −2 and c₂ = −8
∴ $\frac{a_{1}}{a_{2}} = \frac{2}{1}, \frac{b_{1}}{b_{2}} = \frac{3}{- 2} and \frac{c_{1}}{c_{2}} = \frac{- 2}{- 8} = \frac{1}{4}$
∴ $\frac{a_{1}}{a_{2}} \neq \frac{b_{1}}{b_{2}}$
The given system has a unique solution.
Hence, the lines intersect exactly at one point.

Answer:

The correct option is (a).

The given system of equations can be written as follows:
$5 x - 15 y - 8 = 0 and 3 x - 9 y - \frac{24}{5} = 0$
The given equations are of the following form:
a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0
Here, a₁ = 5, b₁ = −15, c₁ = −8 and a₂ = 3, b₂ = −9 and c₂ = $- \frac{24}{5}$
∴ $\frac{a_{1}}{a_{2}} = \frac{5}{3}, \frac{b_{1}}{b_{2}} = \frac{- 15}{- 9} = \frac{5}{3} and \frac{c_{1}}{c_{2}} = - 8 \times \frac{5}{- 24} = \frac{5}{3}$
∴ $\frac{a_{1}}{a_{2}} = \frac{b_{1}}{b_{2}} = \frac{c_{1}}{c_{2}}$
The given system of equations will have an infinite number of solutions.
Hence, the lines are coincident.

Question 1:

The correct option is (a).

The given system of equations can be written as follows:
$5 x - 15 y - 8 = 0 and 3 x - 9 y - \frac{24}{5} = 0$
The given equations are of the following form:
a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0
Here, a₁ = 5, b₁ = −15, c₁ = −8 and a₂ = 3, b₂ = −9 and c₂ = $- \frac{24}{5}$
∴ $\frac{a_{1}}{a_{2}} = \frac{5}{3}, \frac{b_{1}}{b_{2}} = \frac{- 15}{- 9} = \frac{5}{3} and \frac{c_{1}}{c_{2}} = - 8 \times \frac{5}{- 24} = \frac{5}{3}$
∴ $\frac{a_{1}}{a_{2}} = \frac{b_{1}}{b_{2}} = \frac{c_{1}}{c_{2}}$
The given system of equations will have an infinite number of solutions.
Hence, the lines are coincident.

Answer:

The correct option is (a).

The given system of equations can be written as follows:
x + 2y − 3 = 0 and 2x + 4y + 7 = 0
The given equations are of the following form:
a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0
Here, a₁ = 1, b₁ = 2, c₁ = −3 and a₂ = 2, b₂ = 4 and c₂ = 7
∴ $\frac{a_{1}}{a_{2}} = \frac{1}{2}, \frac{b_{1}}{b_{2}} = \frac{2}{4} = \frac{1}{2} and \frac{c_{1}}{c_{2}} = \frac{- 3}{7}$
∴ $\frac{a_{1}}{a_{2}} = \frac{b_{1}}{b_{2}} \neq \frac{c_{1}}{c_{2}}$
So, the given system has no solution.
Hence, the lines are parallel.

Question 2:

The correct option is (a).

The given system of equations can be written as follows:
x + 2y − 3 = 0 and 2x + 4y + 7 = 0
The given equations are of the following form:
a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0
Here, a₁ = 1, b₁ = 2, c₁ = −3 and a₂ = 2, b₂ = 4 and c₂ = 7
∴ $\frac{a_{1}}{a_{2}} = \frac{1}{2}, \frac{b_{1}}{b_{2}} = \frac{2}{4} = \frac{1}{2} and \frac{c_{1}}{c_{2}} = \frac{- 3}{7}$
∴ $\frac{a_{1}}{a_{2}} = \frac{b_{1}}{b_{2}} \neq \frac{c_{1}}{c_{2}}$
So, the given system has no solution.
Hence, the lines are parallel.

Answer:

The correct option is (d).

The given system of equations can be written as follows:
2x − 3y − 7 = 0 and (a + b)x − (a + b − 3)y − (4a + b) = 0
The given equations are of the following form:
a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0
Here, a₁ = 2, b₁ = −3, c₁ = −7 and a₂ = (a + b), b₂ = −(a + b − 3) and c₂ = −(4a + b)
∴ $\frac{a_{1}}{a_{2}} = \frac{2}{(a + b)}, \frac{b_{1}}{b_{2}} = \frac{- 3}{- (a + b - 3)} = \frac{3}{(a + b - 3)} and \frac{c_{1}}{c_{2}} = \frac{- 7}{- (4 a + b)} = \frac{7}{(4 a + b)}$
For an infinite number of solutions, we must have:
$\frac{a_{1}}{a_{2}} = \frac{b_{1}}{b_{2}} = \frac{c_{1}}{c_{2}}$
∴ $\frac{2}{(a + b)} = \frac{3}{(a + b - 3)} = \frac{7}{(4 a + b)}$
Now, we have:
$\frac{2}{(a + b)} = \frac{3}{(a + b - 3)} \Rightarrow 2 a + 2 b - 6 = 3 a + 3 b$
⇒ a + b + 6 = 0                    ...(i)
Again, we have:
$\frac{3}{(a + b - 3)} = \frac{7}{(4 a + b)} \Rightarrow 12 a + 3 b = 7 a + 7 b - 21$
⇒ 5a − 4b + 21 = 0                            ...(ii)

On multiplying (i) by 4, we get:
4a + 4b + 24 = 0                              ...(iii)
On adding (ii) and (iii), we get:
9a = −45 ⇒ a = −5
On substituting a = −5 in (i), we get:
−5 + b + 6 = 0 ⇒ b = −1
∴ â€‹a = −5 and b = −1

Question 3:

The correct option is (d).

The given system of equations can be written as follows:
2x − 3y − 7 = 0 and (a + b)x − (a + b − 3)y − (4a + b) = 0
The given equations are of the following form:
a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0
Here, a₁ = 2, b₁ = −3, c₁ = −7 and a₂ = (a + b), b₂ = −(a + b − 3) and c₂ = −(4a + b)
∴ $\frac{a_{1}}{a_{2}} = \frac{2}{(a + b)}, \frac{b_{1}}{b_{2}} = \frac{- 3}{- (a + b - 3)} = \frac{3}{(a + b - 3)} and \frac{c_{1}}{c_{2}} = \frac{- 7}{- (4 a + b)} = \frac{7}{(4 a + b)}$
For an infinite number of solutions, we must have:
$\frac{a_{1}}{a_{2}} = \frac{b_{1}}{b_{2}} = \frac{c_{1}}{c_{2}}$
∴ $\frac{2}{(a + b)} = \frac{3}{(a + b - 3)} = \frac{7}{(4 a + b)}$
Now, we have:
$\frac{2}{(a + b)} = \frac{3}{(a + b - 3)} \Rightarrow 2 a + 2 b - 6 = 3 a + 3 b$
⇒ a + b + 6 = 0                    ...(i)
Again, we have:
$\frac{3}{(a + b - 3)} = \frac{7}{(4 a + b)} \Rightarrow 12 a + 3 b = 7 a + 7 b - 21$
⇒ 5a − 4b + 21 = 0                            ...(ii)

On multiplying (i) by 4, we get:
4a + 4b + 24 = 0                              ...(iii)
On adding (ii) and (iii), we get:
9a = −45 ⇒ a = −5
On substituting a = −5 in (i), we get:
−5 + b + 6 = 0 ⇒ b = −1
∴ â€‹a = −5 and b = −1

Answer:

The correct option is (a).

The given system of equations can be written as follows:
2x + y − 5 = 0 and 3x + 2y − 8 = 0
The given equations are of the following form:
a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0
Here, a₁ = 2, b₁ = 1, c₁ = −5 and a₂ = 3, b₂ = 2 and c₂ = −8
∴ $\frac{a_{1}}{a_{2}} = \frac{2}{3}, \frac{b_{1}}{b_{2}} = \frac{1}{2} and \frac{c_{1}}{c_{2}} = \frac{- 5}{- 8} = \frac{5}{8}$
∴ $\frac{a_{1}}{a_{2}} \neq \frac{b_{1}}{b_{2}}$
The given system has a unique solution.
Hence, the lines intersect at one point.

Question 4:

The correct option is (a).

The given system of equations can be written as follows:
2x + y − 5 = 0 and 3x + 2y − 8 = 0
The given equations are of the following form:
a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0
Here, a₁ = 2, b₁ = 1, c₁ = −5 and a₂ = 3, b₂ = 2 and c₂ = −8
∴ $\frac{a_{1}}{a_{2}} = \frac{2}{3}, \frac{b_{1}}{b_{2}} = \frac{1}{2} and \frac{c_{1}}{c_{2}} = \frac{- 5}{- 8} = \frac{5}{8}$
∴ $\frac{a_{1}}{a_{2}} \neq \frac{b_{1}}{b_{2}}$
The given system has a unique solution.
Hence, the lines intersect at one point.

Answer:

The correct option is (d).

Given:
x = −y and y > 0
Now, we have:
(i) x²y
On substituting x = −y, we get:
(−y)²y = y³ > 0 (âˆµ y > 0)
This is true.

(ii) x + y
On substituting x = −y, we get:
(−y) + y = 0
This is also true.

(iii) xy
On substituting x = −y, we get:
(−y) y = −y² < 0 (âˆµ y > 0)
This is again true.

(iv) $\frac{1}{x} - \frac{1}{y} = 0$
$\Rightarrow \frac{y - x}{x y} = 0$
On substituting x = −y, we get:
$\frac{y - (- y)}{(- y) y} = 0 \Rightarrow \frac{2 y}{- y^{2}} = 0 \Rightarrow 2 y = 0 \Rightarrow y = 0$
Hence, from the above equation, we get y = 0, which is wrong.

Question 5:

The correct option is (d).

Given:
x = −y and y > 0
Now, we have:
(i) x²y
On substituting x = −y, we get:
(−y)²y = y³ > 0 (âˆµ y > 0)
This is true.

(ii) x + y
On substituting x = −y, we get:
(−y) + y = 0
This is also true.

(iii) xy
On substituting x = −y, we get:
(−y) y = −y² < 0 (âˆµ y > 0)
This is again true.

(iv) $\frac{1}{x} - \frac{1}{y} = 0$
$\Rightarrow \frac{y - x}{x y} = 0$
On substituting x = −y, we get:
$\frac{y - (- y)}{(- y) y} = 0 \Rightarrow \frac{2 y}{- y^{2}} = 0 \Rightarrow 2 y = 0 \Rightarrow y = 0$
Hence, from the above equation, we get y = 0, which is wrong.

Answer:

The given system of equations:
$- x + 2 y + 2 = 0 and \frac{1}{2} x - \frac{1}{4} y - 1 = 0$
The given equations are of the following form:
a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0
Here, a₁ = −1, b₁ = 2, c₁ = 2 and a₂ = $\frac{1}{2}$ , b₂ = $- \frac{1}{4}$ and c₂ = −1
∴ $\frac{a_{1}}{a_{2}} = \frac{- 1}{(\frac{1}{2})} = - 2, \frac{b_{1}}{b_{2}} = \frac{2}{(- \frac{1}{4})} = - 8 and \frac{c_{1}}{c_{2}} = \frac{2}{- 1} = - 2$
∴ $\frac{a_{1}}{a_{2}} \neq \frac{b_{1}}{b_{2}}$
The given system has a unique solution.
Hence, the lines intersect at one point.

Question 6:

The given system of equations:
$- x + 2 y + 2 = 0 and \frac{1}{2} x - \frac{1}{4} y - 1 = 0$
The given equations are of the following form:
a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0
Here, a₁ = −1, b₁ = 2, c₁ = 2 and a₂ = $\frac{1}{2}$ , b₂ = $- \frac{1}{4}$ and c₂ = −1
∴ $\frac{a_{1}}{a_{2}} = \frac{- 1}{(\frac{1}{2})} = - 2, \frac{b_{1}}{b_{2}} = \frac{2}{(- \frac{1}{4})} = - 8 and \frac{c_{1}}{c_{2}} = \frac{2}{- 1} = - 2$
∴ $\frac{a_{1}}{a_{2}} \neq \frac{b_{1}}{b_{2}}$
The given system has a unique solution.
Hence, the lines intersect at one point.

Answer:

The given system of equations can be written as follows:
kx + 3y − (k − 2) = 0 and 12x + ky − k = 0
The given equations are of the following form:
a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0
Here, a₁ = k, b₁ = 3, c₁ = − (k − 2) and a₂ = 12, b₂ = k and c₂ = − k
∴ $\frac{a_{1}}{a_{2}} = \frac{k}{12}, \frac{b_{1}}{b_{2}} = \frac{3}{k} and \frac{c_{1}}{c_{2}} = \frac{- (k - 2)}{- k} = \frac{(k - 2)}{k}$
For inconsistency, we must have:
$\frac{a_{1}}{a_{2}} = \frac{b_{1}}{b_{2}} \neq \frac{c_{1}}{c_{2}}$
⇒ $\frac{k}{12} = \frac{3}{k} \neq \frac{(k - 2)}{k} \Rightarrow k^{2} = (3 \times 12) = 36$
⇒ $k = \sqrt{36} = \pm 6$
Hence, the pair of equations is inconsistent if $k = \pm 6$ .

Question 7:

The given system of equations can be written as follows:
kx + 3y − (k − 2) = 0 and 12x + ky − k = 0
The given equations are of the following form:
a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0
Here, a₁ = k, b₁ = 3, c₁ = − (k − 2) and a₂ = 12, b₂ = k and c₂ = − k
∴ $\frac{a_{1}}{a_{2}} = \frac{k}{12}, \frac{b_{1}}{b_{2}} = \frac{3}{k} and \frac{c_{1}}{c_{2}} = \frac{- (k - 2)}{- k} = \frac{(k - 2)}{k}$
For inconsistency, we must have:
$\frac{a_{1}}{a_{2}} = \frac{b_{1}}{b_{2}} \neq \frac{c_{1}}{c_{2}}$
⇒ $\frac{k}{12} = \frac{3}{k} \neq \frac{(k - 2)}{k} \Rightarrow k^{2} = (3 \times 12) = 36$
⇒ $k = \sqrt{36} = \pm 6$
Hence, the pair of equations is inconsistent if $k = \pm 6$ .

Answer:

The given system of equations can be written as follows:
$9 x - 10 y - 21 = 0 and \frac{3 x}{2} - \frac{5 y}{3} - \frac{7}{2} = 0$
The given equations are of the following form:
a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0
Here, a₁ = 9, b₁ = −10, c₁ = −21 and a₂ = $\frac{3}{2}$ , b₂ = $\frac{- 5}{3}$ and c₂ = $\frac{- 7}{2}$
∴ $\frac{a_{1}}{a_{2}} = \frac{9}{\frac{3}{2}} = 6, \frac{b_{1}}{b_{2}} = \frac{- 10}{(\frac{- 5}{3})} = 6 and \frac{c_{1}}{c_{2}} = - 21 \times \frac{2}{- 7} = 6$
∴ $\frac{a_{1}}{a_{2}} = \frac{b_{1}}{b_{2}} = \frac{c_{1}}{c_{2}}$
This shows that the given system equations has an infinite number of solutions.

Question 8:

The given system of equations can be written as follows:
$9 x - 10 y - 21 = 0 and \frac{3 x}{2} - \frac{5 y}{3} - \frac{7}{2} = 0$
The given equations are of the following form:
a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0
Here, a₁ = 9, b₁ = −10, c₁ = −21 and a₂ = $\frac{3}{2}$ , b₂ = $\frac{- 5}{3}$ and c₂ = $\frac{- 7}{2}$
∴ $\frac{a_{1}}{a_{2}} = \frac{9}{\frac{3}{2}} = 6, \frac{b_{1}}{b_{2}} = \frac{- 10}{(\frac{- 5}{3})} = 6 and \frac{c_{1}}{c_{2}} = - 21 \times \frac{2}{- 7} = 6$
∴ $\frac{a_{1}}{a_{2}} = \frac{b_{1}}{b_{2}} = \frac{c_{1}}{c_{2}}$
This shows that the given system equations has an infinite number of solutions.

Answer:

The given equations are as follows:
x − 2y = 0                     ....(i)
3x + 4y = 20                 ....(ii)
On multiplying (i) by 2, we get:
2x − 4y = 0                        ....(iii)
On adding (ii) and (iii), we get:
5x = 20 ⇒ x = 4
On substituting x = 4 in (i), we get:
4 − 2y = 0 ⇒ 4 = 2y ⇒ y = 2
Hence, the required solution is x = 4 and y = 2.

Question 9:

The given equations are as follows:
x − 2y = 0                     ....(i)
3x + 4y = 20                 ....(ii)
On multiplying (i) by 2, we get:
2x − 4y = 0                        ....(iii)
On adding (ii) and (iii), we get:
5x = 20 ⇒ x = 4
On substituting x = 4 in (i), we get:
4 − 2y = 0 ⇒ 4 = 2y ⇒ y = 2
Hence, the required solution is x = 4 and y = 2.

Answer:

The given system of equations can be written as follows:
x − 3y − 2 = 0 and −2x + 6y − 5= 0
The given equations are of the following form:
a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0
Here, a₁ = 1, b₁ = −3, c₁ = −2 and a₂ = −2, b₂ = 6 and c₂ = −5
∴ $\frac{a_{1}}{a_{2}} = \frac{1}{- 2} = \frac{- 1}{2}, \frac{b_{1}}{b_{2}} = \frac{- 3}{6} = \frac{- 1}{2} and \frac{c_{1}}{c_{2}} = \frac{- 2}{- 5} = \frac{2}{5}$
∴ â€‹ $\frac{a_{1}}{a_{2}} = \frac{b_{1}}{b_{2}} \neq \frac{c_{1}}{c_{2}}$
Thus, the given system of equations has no solution.
Hence, the paths represented by the equations are parallel.

Question 10:

The given system of equations can be written as follows:
x − 3y − 2 = 0 and −2x + 6y − 5= 0
The given equations are of the following form:
a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0
Here, a₁ = 1, b₁ = −3, c₁ = −2 and a₂ = −2, b₂ = 6 and c₂ = −5
∴ $\frac{a_{1}}{a_{2}} = \frac{1}{- 2} = \frac{- 1}{2}, \frac{b_{1}}{b_{2}} = \frac{- 3}{6} = \frac{- 1}{2} and \frac{c_{1}}{c_{2}} = \frac{- 2}{- 5} = \frac{2}{5}$
∴ â€‹ $\frac{a_{1}}{a_{2}} = \frac{b_{1}}{b_{2}} \neq \frac{c_{1}}{c_{2}}$
Thus, the given system of equations has no solution.
Hence, the paths represented by the equations are parallel.

Answer:

Let the larger number be x and the smaller number be y.
Then, we have:
x − y = 26 ...(i)
x = 3y ...(ii)
On substituting x = 3y in (i), we get:
3y − y = 26 ⇒ 2y = 26 ⇒ y = 13
On substituting y = 13 in (i), we get:
x − 13 = 26 ⇒ x = 26 + 13 = 39
Hence, the required numbers are 39 and 13.

Question 11:

Let the larger number be x and the smaller number be y.
Then, we have:
x − y = 26 ...(i)
x = 3y ...(ii)
On substituting x = 3y in (i), we get:
3y − y = 26 ⇒ 2y = 26 ⇒ y = 13
On substituting y = 13 in (i), we get:
x − 13 = 26 ⇒ x = 26 + 13 = 39
Hence, the required numbers are 39 and 13.

Answer:

The given equations are as follows:
23x + 29y = 98                              ....(i)
29x + 23y = 110                          ....(ii)
On adding (i) and (ii), we get:
52x + 52y = 208
⇒ x + y = 4                           ....(iii)
On subtracting (i) from (ii), we get:
6x − 6y = 12
⇒ x − y = 2                       ....(iv)
On adding (iii) and (iv), we get:
2x = 6 ⇒ x = 3
On substituting x = 3 in (iii), we get:
3 + y = 4
⇒ y = 4 − 3 = 1
Hence, the required solution is x = 3 and y = 1.

Question 12:

The given equations are as follows:
23x + 29y = 98                              ....(i)
29x + 23y = 110                          ....(ii)
On adding (i) and (ii), we get:
52x + 52y = 208
⇒ x + y = 4                           ....(iii)
On subtracting (i) from (ii), we get:
6x − 6y = 12
⇒ x − y = 2                       ....(iv)
On adding (iii) and (iv), we get:
2x = 6 ⇒ x = 3
On substituting x = 3 in (iii), we get:
3 + y = 4
⇒ y = 4 − 3 = 1
Hence, the required solution is x = 3 and y = 1.

Answer:

The given equations are as follows:
6x + 3y = 7xy                   ....(i)
3x + 9y = 11xy                     ....(ii)

For equation (i), we have:

$\frac{6 x + 3 y}{x y} = 7$
$\Rightarrow \frac{6 x}{x y} + \frac{3 y}{x y} = 7 \Rightarrow \frac{6}{y} + \frac{3}{x} = 7$       ....(iii)

For equation (ii), we have:

$\frac{3 x + 9 y}{x y} = 11$
$\Rightarrow \frac{3 x}{x y} + \frac{9 y}{x y} = 11 \Rightarrow \frac{3}{y} + \frac{9}{x} = 11$       ....(iv)
On substituting $\frac{1}{y} = v$ and $\frac{1}{x} = u$ in (iii) and (iv), we get:
6v + 3u = 7                              ....(v)
3v + 9u = 11                            ....(vi)
On multiplying (v) by 3, we get:
18v + 9u = 21                          ....(vii)
On subtracting (vi) from (vii), we get:
15v = 10 ⇒ v = $\frac{10}{15} = \frac{2}{3}$
⇒ $\frac{1}{y} = \frac{2}{3} \Rightarrow y = \frac{3}{2}$
On substituting $y = \frac{3}{2}$ in (iii), we get:
$\frac{6}{(\frac{3}{2})} + \frac{3}{x} = 7$
$\Rightarrow 4 + \frac{3}{x} = 7 \Rightarrow \frac{3}{x} = 3 \Rightarrow 3 x = 3$
$\Rightarrow x = 1$
Hence, the required solution is x = 1 and $y = \frac{3}{2}$ .

Question 13:

The given equations are as follows:
6x + 3y = 7xy                   ....(i)
3x + 9y = 11xy                     ....(ii)

For equation (i), we have:

$\frac{6 x + 3 y}{x y} = 7$
$\Rightarrow \frac{6 x}{x y} + \frac{3 y}{x y} = 7 \Rightarrow \frac{6}{y} + \frac{3}{x} = 7$       ....(iii)

For equation (ii), we have:

$\frac{3 x + 9 y}{x y} = 11$
$\Rightarrow \frac{3 x}{x y} + \frac{9 y}{x y} = 11 \Rightarrow \frac{3}{y} + \frac{9}{x} = 11$       ....(iv)
On substituting $\frac{1}{y} = v$ and $\frac{1}{x} = u$ in (iii) and (iv), we get:
6v + 3u = 7                              ....(v)
3v + 9u = 11                            ....(vi)
On multiplying (v) by 3, we get:
18v + 9u = 21                          ....(vii)
On subtracting (vi) from (vii), we get:
15v = 10 ⇒ v = $\frac{10}{15} = \frac{2}{3}$
⇒ $\frac{1}{y} = \frac{2}{3} \Rightarrow y = \frac{3}{2}$
On substituting $y = \frac{3}{2}$ in (iii), we get:
$\frac{6}{(\frac{3}{2})} + \frac{3}{x} = 7$
$\Rightarrow 4 + \frac{3}{x} = 7 \Rightarrow \frac{3}{x} = 3 \Rightarrow 3 x = 3$
$\Rightarrow x = 1$
Hence, the required solution is x = 1 and $y = \frac{3}{2}$ .

Answer:

The given system of equations:
3x + y = 1
⇒ 3x + y − 1= 0 ....(i)
kx + 2y = 5
⇒ kx + 2y − 5 = 0 ....(ii)
These equations are of the following form:
a₁x + b₁y + c₁= 0, a₂x + b₂y + c₂ = 0
Here, a₁ = 3, b₁= 1, c₁ = −1 and a₂ = k, b₂= 2, c₂ = −5

(i) For a unique solution, we must have:
$\frac{a_{1}}{a_{2}} \neq \frac{b_{1}}{b_{2}}$ i.e. $\frac{3}{k} \neq \frac{1}{2} \Rightarrow k \neq 6$
Thus, for all real values of k other than 6, the given system of equations will have a unique solution.

(ii) In order that the given equations have no solution, we must have:
$\frac{a_{1}}{a_{2}} = \frac{b_{1}}{b_{2}} \neq \frac{c_{1}}{c_{2}}$
$\frac{3}{k} = \frac{1}{2} \neq \frac{- 1}{- 5}$
$\Rightarrow \frac{3}{k} = \frac{1}{2} and \frac{3}{k} \neq \frac{- 1}{- 5}$
$\Rightarrow k = 6, k \neq 15$
Thus, for k = 6, the given system of equations will have no solution.

Question 14:

The given system of equations:
3x + y = 1
⇒ 3x + y − 1= 0 ....(i)
kx + 2y = 5
⇒ kx + 2y − 5 = 0 ....(ii)
These equations are of the following form:
a₁x + b₁y + c₁= 0, a₂x + b₂y + c₂ = 0
Here, a₁ = 3, b₁= 1, c₁ = −1 and a₂ = k, b₂= 2, c₂ = −5

(i) For a unique solution, we must have:
$\frac{a_{1}}{a_{2}} \neq \frac{b_{1}}{b_{2}}$ i.e. $\frac{3}{k} \neq \frac{1}{2} \Rightarrow k \neq 6$
Thus, for all real values of k other than 6, the given system of equations will have a unique solution.

(ii) In order that the given equations have no solution, we must have:
$\frac{a_{1}}{a_{2}} = \frac{b_{1}}{b_{2}} \neq \frac{c_{1}}{c_{2}}$
$\frac{3}{k} = \frac{1}{2} \neq \frac{- 1}{- 5}$
$\Rightarrow \frac{3}{k} = \frac{1}{2} and \frac{3}{k} \neq \frac{- 1}{- 5}$
$\Rightarrow k = 6, k \neq 15$
Thus, for k = 6, the given system of equations will have no solution.

Answer:

Let $∠ A = x ° and ∠ B = y °$
Then, $∠ C = 3 ∠ B = (3 y) °$
Now, we have:
$∠ A + ∠ B + ∠ C = 180 °$
⇒ x + y + 3y = 180
⇒ x + 4y = 180              ....(i)
Also, $∠ C = 2 (∠ A + ∠ B)$
⇒ 3y = 2(x + y)
⇒ 2x − y = 0                ....(ii)
On multiplying (ii) by 4, we get:
8x − 4y = 0                    ....(iii)
On adding (i) and (iii), we get:
9x = 180 ⇒ x = 20
On substituting x = 20 in (i), we get:
20 + 4y = 180 ⇒ 4y = (180 − 20) = 160 ⇒ y = 40
∴ x = 20 and y = 40
∴ â€‹ $∠ A = 20 °, ∠ B = 40 °, ∠ C = (3 \times 40) ° = 120 °$

Question 15:

Let $∠ A = x ° and ∠ B = y °$
Then, $∠ C = 3 ∠ B = (3 y) °$
Now, we have:
$∠ A + ∠ B + ∠ C = 180 °$
⇒ x + y + 3y = 180
⇒ x + 4y = 180              ....(i)
Also, $∠ C = 2 (∠ A + ∠ B)$
⇒ 3y = 2(x + y)
⇒ 2x − y = 0                ....(ii)
On multiplying (ii) by 4, we get:
8x − 4y = 0                    ....(iii)
On adding (i) and (iii), we get:
9x = 180 ⇒ x = 20
On substituting x = 20 in (i), we get:
20 + 4y = 180 ⇒ 4y = (180 − 20) = 160 ⇒ y = 40
∴ x = 20 and y = 40
∴ â€‹ $∠ A = 20 °, ∠ B = 40 °, ∠ C = (3 \times 40) ° = 120 °$

Answer:

Let the cost of each pencil be Rs. x and that of each pen be Rs. y.
Then, we have:
5x + 7y = 195                       ....(i)
7x + 5y = 153                   ....(ii)
Adding (i) and (ii), we get:
12x + 12y = 348
⇒ 12(x + y) = 348
⇒ x + y = 29                             ....(iii)
Subtracting (i) from (ii), we get:
2x − 2y = −42
⇒ 2(x − y) = −42
⇒ x − y = −21                          ....(iv)
On adding (iii) and (iv), we get:
2x = 8 ⇒ x = 4
On substituting x = 4 in (iii), we get:
4 + y = 29 ⇒ y = (29 − 4) = 25
Hence, the cost of each pencil is Rs. 4 and the cost of each pen is Rs. 25.

Question 16:

Let the cost of each pencil be Rs. x and that of each pen be Rs. y.
Then, we have:
5x + 7y = 195                       ....(i)
7x + 5y = 153                   ....(ii)
Adding (i) and (ii), we get:
12x + 12y = 348
⇒ 12(x + y) = 348
⇒ x + y = 29                             ....(iii)
Subtracting (i) from (ii), we get:
2x − 2y = −42
⇒ 2(x − y) = −42
⇒ x − y = −21                          ....(iv)
On adding (iii) and (iv), we get:
2x = 8 ⇒ x = 4
On substituting x = 4 in (iii), we get:
4 + y = 29 ⇒ y = (29 − 4) = 25
Hence, the cost of each pencil is Rs. 4 and the cost of each pen is Rs. 25.

Answer:

On a graph paper, draw a horizontal line X'OX and a vertical line YOY' as the x-axis and the y-axis, respectively.

Graph of 2x − 3y = 1
2x − 3y = 1
⇒ 3y = (2x − 1)
∴ $y = \frac{2 x - 1}{3}$ ...(i)
Putting x = −1, we get:
y = −1
Putting x = 2, we get:
y = 1
Putting x = 5, we get:
y = 3
Thus, we have the following table for the equation 2x − 3y = 1.

x	−1	2	5
y	−1	1	3

Now, plots the points A(−1, −1), B(2, 1) and C(5, 3) on the graph paper.
Join AB and BC to get the graph line AC. Extend it on both the sides.
Thus, the line AC is the graph of 2x − 3y = 1.

Graph of 4x − 3y + 1 = 0
4x − 3y + 1 = 0
⇒ 3y = ( 4x + 1)
∴

y = \frac{4 x + 1}{3}

...(ii)
Putting x = −1, we get:
y = −1
Putting x = 2, we get:
y = 3
Putting x = 5, we get:
y = 7
Thus, we have the following table for the equation 4x − 3y + 1 = 0.

x	−1	2	5
y	−1	3	7

Now, plots the points P(2, 3) and Q(5, 7). The point A(−1, −1) has already been plotted. Join PA and QP to get the graph line AQ. Extend it on both the sides.
Thus, the line AQ is the graph of the equation 4x − 3y + 1 = 0.

The two lines intersect at A(−1, −1).
Thus, x = −1 and y = −1 is the solution of the given system of equations.

Question 17:

On a graph paper, draw a horizontal line X'OX and a vertical line YOY' as the x-axis and the y-axis, respectively.

Graph of 2x − 3y = 1
2x − 3y = 1
⇒ 3y = (2x − 1)
∴ $y = \frac{2 x - 1}{3}$ ...(i)
Putting x = −1, we get:
y = −1
Putting x = 2, we get:
y = 1
Putting x = 5, we get:
y = 3
Thus, we have the following table for the equation 2x − 3y = 1.

x	−1	2	5
y	−1	1	3

Now, plots the points A(−1, −1), B(2, 1) and C(5, 3) on the graph paper.
Join AB and BC to get the graph line AC. Extend it on both the sides.
Thus, the line AC is the graph of 2x − 3y = 1.

Graph of 4x − 3y + 1 = 0
4x − 3y + 1 = 0
⇒ 3y = ( 4x + 1)
∴

y = \frac{4 x + 1}{3}

...(ii)
Putting x = −1, we get:
y = −1
Putting x = 2, we get:
y = 3
Putting x = 5, we get:
y = 7
Thus, we have the following table for the equation 4x − 3y + 1 = 0.

x	−1	2	5
y	−1	3	7

Now, plots the points P(2, 3) and Q(5, 7). The point A(−1, −1) has already been plotted. Join PA and QP to get the graph line AQ. Extend it on both the sides.
Thus, the line AQ is the graph of the equation 4x − 3y + 1 = 0.

The two lines intersect at A(−1, −1).
Thus, x = −1 and y = −1 is the solution of the given system of equations.

Answer:

Given:
In a cyclic quadrilateral ABCD, we have:
$∠ A = (4 x + 20) °$
$∠ B = (3 x - 5) °$
$∠ C = (4 y) °$
$∠ D = (7 y + 5) °$
$∠ A + ∠ C = 180 °$ and $∠ B + ∠ D = 180 °$       [Since ABCD is a cyclic quadrilateral]
Now, $∠ A + ∠ C = (4 x + 20) ° + (4 y) ° = 180 °$
⇒ 4x + 4y + 20 = 180
⇒ 4x + 4y = 180 − 20 = 160
⇒ x + y = 40                       ....(i)
Also, $∠ B + ∠ D = (3 x - 5) ° + (7 y + 5) ° = 180 °$
⇒ 3x + 7y = 180                 ....(ii)
On multiplying (i) by 3, we get:
3x + 3y = 120                     ....(iii)
On subtracting (iii) from (ii), we get:
4y = 60 ⇒ y = 15
On substituting y = 15 in (1), we get:
x + 15 = 40 ⇒ x = (40 − 15) = 25
Therefore, we have:
$∠ A = (4 x + 20) ° = (4 \times 25 + 20) ° = 120 °$
$∠ B = (3 x - 5) ° = (3 \times 25 - 5) ° = 70 °$
$∠ C = (4 y) ° = (4 \times 15) ° = 60 °$
$∠ D = (7 y + 5) ° = (7 \times 15 + 5) ° = (105 + 5) ° = 110 °$

Question 18:

Given:
In a cyclic quadrilateral ABCD, we have:
$∠ A = (4 x + 20) °$
$∠ B = (3 x - 5) °$
$∠ C = (4 y) °$
$∠ D = (7 y + 5) °$
$∠ A + ∠ C = 180 °$ and $∠ B + ∠ D = 180 °$       [Since ABCD is a cyclic quadrilateral]
Now, $∠ A + ∠ C = (4 x + 20) ° + (4 y) ° = 180 °$
⇒ 4x + 4y + 20 = 180
⇒ 4x + 4y = 180 − 20 = 160
⇒ x + y = 40                       ....(i)
Also, $∠ B + ∠ D = (3 x - 5) ° + (7 y + 5) ° = 180 °$
⇒ 3x + 7y = 180                 ....(ii)
On multiplying (i) by 3, we get:
3x + 3y = 120                     ....(iii)
On subtracting (iii) from (ii), we get:
4y = 60 ⇒ y = 15
On substituting y = 15 in (1), we get:
x + 15 = 40 ⇒ x = (40 − 15) = 25
Therefore, we have:
$∠ A = (4 x + 20) ° = (4 \times 25 + 20) ° = 120 °$
$∠ B = (3 x - 5) ° = (3 \times 25 - 5) ° = 70 °$
$∠ C = (4 y) ° = (4 \times 15) ° = 60 °$
$∠ D = (7 y + 5) ° = (7 \times 15 + 5) ° = (105 + 5) ° = 110 °$

Answer:

We have:
$\frac{35}{x + y} + \frac{14}{x - y} = 19 and \frac{14}{x + y} + \frac{35}{x - y} = 37$
Taking $\frac{1}{x + y} = u$ and $\frac{1}{x - y} = v$ :
35u + 14v − 19 = 0                    ....(i)
14u + 35v − 37 = 0                    ....(ii)
Here, a₁= 35, b₁ = 14, c₁ = −19, a₂ = 14, b₂ = 35, c₂ = −37
By cross multiplication, we have:

∴ $\frac{u}{[14 \times (- 37) - 35 \times (- 19)]} = \frac{v}{[(- 19) \times 14 - (- 37) \times (35)]} = \frac{1}{[35 \times 35 - 14 \times 14]}$
⇒ $\frac{u}{- 518 + 665} = \frac{v}{- 266 + 1295} = \frac{1}{1225 - 196}$
⇒ $\frac{u}{147} = \frac{v}{1029} = \frac{1}{1029}$
⇒ $u = \frac{147}{1029} = \frac{1}{7}, v = \frac{1029}{1029} = 1$
⇒ $\frac{1}{x + y} = \frac{1}{7}, \frac{1}{x - y} = 1$
∴ (x + y) = 7                          ....(iii)
And, (x − y) = 1 ....(iv)

Again, the equations (iii) and (iv) can be written as follows:
x + y − 7 = 0                               ....(v)
x − y − 1 = 0 ....(vi)
Here, a₁= 1, b₁ = 1, c₁ = −7 , a₂ = 1 , b₂ = −1 , c₂ = −1
By cross multiplication, we have:

⇒ $\frac{x}{[1 \times (- 1) - (- 1) \times (- 7)]} = \frac{y}{[(- 7) \times 1 - (- 1) \times 1]} = \frac{1}{[1 \times (- 1) - 1 \times 1]}$

⇒ $\frac{x}{- 1 - 7} = \frac{y}{- 7 + 1} = \frac{1}{- 1 - 1}$
⇒ $\frac{x}{- 8} = \frac{y}{- 6} = \frac{1}{- 2}$
⇒ $x = \frac{- 8}{- 2} = 4, y = \frac{- 6}{- 2} = 3$
Hence, x = 4 and y = 3 is the required solution.

RS Aggarwal 2021 2022 Solutions for Class 10 Maths Chapter 3 - Linear Equations In Two Variables

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