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RD Sharma (2014) Solutions for Class 7 Math Chapter 7 - Algebraic Expressions

RD Sharma (2014) Solutions for Class 7 Math Chapter 7 Algebraic Expressions are provided here with simple step-by-step explanations. These solutions for Algebraic Expressions are extremely popular among class 7 students for Math Algebraic Expressions Solutions come handy for quickly completing your homework and preparing for exams. All questions and answers from the RD Sharma (2014) Book of class 7 Math Chapter 7 are provided here for you for free. You will also love the ad-free experience on Meritnation’s RD Sharma (2014) Solutions. All RD Sharma (2014) Solutions for class 7 Math are prepared by experts and are 100% accurate.

Page No 7.13:

Question 1:

Add the following:
(i) 3x and 7x
(ii) −5xy and 9xy

Answer:

We have
(i) 3x + 7x = (3 + 7)x = 10x
(ii) -5xy + 9xy = ( -5 + 9)xy = 4xy

Page No 7.13:

Question 2:

Simplify each of the following:
(i) 7x3y + 9yx3
(ii) 12a2b + 3ba2

Answer:

Simplifying the given expressions, we have
(i) 7x3y + 9yx3 = (7 + 9)x3y = 16x3y
(ii) 12a2b + 3ba2 = (12 + 3)a2b = 15a2b

Page No 7.13:

Question 3:

Add the following:
(i) 7abc, −5abc, 9abc, −8abc
(ii) 2x2y, − 4x2y, 6x2y, −5x2y

Answer:

Adding the given terms, we have
(i) 7abc + (- 5abc) + (9 abc) + (- 8abc)
    = 7abc - 5abc + 9abc - 8abc
    = (7 - 5 + 9 - 8)abc
    = (16 - 13)abc
    = 3abc

(ii) 2x2y + (- 4x2y) + 6x2y + (- 5x2y)
    = 2x2y - 4x2y + 6x2y - 5x2y
    = (2 - 4 + 6 - 5)x2y
    = (8 - 9)x2y
    = -x2y



Page No 7.14:

Question 4:

Add the following expressions:
(i) x3-2x2y+3xy2-y3, 2x3-5xy2+3x2y-4y3
(ii) a4-2a3b+3ab3+4a2b2+3b4,-2a4-5ab3+7a3b-6a2b2+b4

Answer:

Adding the given expressions, we have
(i) x3- 2x2y + 3xy2- y3+ 2x3- 5xy2 + 3x2y- 4y3
     Collecting positive and negative like terms together, we get
     x3+ 2x3- 2x2y + 3x2y + 3xy2- 5xy2 - y3 - 4y3
   = 3x3 + x2y - 2xy2 - 5y3

(ii) (a4- 2a3b + 3ab3 + 4a2b2 + 3b4) + (-2a4- 5ab3 + 7a3b - 6a2b2 + b4)
      a4- 2a3b + 3ab3 + 4a2b2 + 3b4 - 2a4- 5ab3 + 7a3b - 6a2b2 + b4
      Collecting positive and negative like terms together, we get
     a4 - 2a4 - 2a3b + 7a3b + 3ab3 - 5ab3 + 4a2b2 - 6a2b2 + 3b4 + b4
    = - a4 + 5a3b - 2ab3 -  2a2b2 + 4b4

Page No 7.14:

Question 5:

Add the following expressions:
(i) 8a-6ab+5b, -6a-ab-8b and -4a+2ab+3b
(ii) 5x3+7+6x-5x2, 2x2-8-9x, 4x-2x2+3x3, 3x3-9x-x2 and x-x2-x3-4

Answer:

(i) Required expression = (8a - 6ab + 5b) + (- 6a - ab - 8b) + ( - 4a + 2ab + 3b)
     Collecting positive and negative like terms together, we get
     8a - 6a - 4a - 6ab - ab + 2ab + 5b - 8b + 3b
     = 8a - 10a - 7ab + 2ab + 8b - 8b
     = - 2a - 5ab

(ii) Required expression = (5x3 + 7 + 6x - 5x2) + (2x2 - 8 - 9x) + (4x - 2x2 + 3x3) + (3x3- 9x - x2) + ( x - x2 - x3- 4)
      Collecting positive and negative like terms together, we get
      5x3+ 3x3 + 3x3- x3- 5x2 + 2x2 - 2x2 - x2- x2 + 6x - 9x + 4x - 9x + x + 7 - 8 - 4
    = 11x3 - x3 - 7x2 + 11x - 18x + 7 - 12
    = 10x3 - 7x2 - 7x - 5

Page No 7.14:

Question 6:

Add the following:
(i) x-3y-2z5x+7y-8z3x-2y+5z
(ii) 4ab-5bc+7ca-3ab+2bc-3ca5ab-3bc+4ca

Answer:

(i)  Required expression = (x - 3y - 2z) + (5x +7y - 8z) +(3x - 2y + 5z)
     Collecting positive and negative like terms together, we get
     x + 5x + 3x - 3y + 7y - 2y - 2z - 8z + 5z
  = 9x - 5y + 7y - 10z + 5z
  = 9x + 2y - 5z

(ii) Required expression = (4ab - 5bc + 7ca) + (- 3ab + 2bc - 3ca ) + (5ab - 3bc + 4ca)
      Collecting positive and negative like terms together, we get
      4ab - 3ab + 5ab - 5bc + 2bc - 3bc + 7ca - 3ca + 4ca
   = 9ab - 3ab - 8bc + 2bc + 11 ca  - 3ca
   = 6ab - 6bc + 8ca

Page No 7.14:

Question 7:

Add 2x2 − 3x + 1 to the sum of 3x2 − 2x and 3x + 7.

Answer:

 Sum of 3x2 - 2x and 3x + 7
= (3x2 - 2x) + ( 3x +7)
= 3x2 - 2x + 3x + 7
= (3x2 + x  + 7)
Now, required expression = (2x2 - 3x + 1) + (3x2 + x  + 7)
                                            = 2x2 + 3x2 - 3x + x + 1 + 7
                                            = 5x2 - 2x + 8

Page No 7.14:

Question 8:

Add x2 + 2xy + y2 to the sum of x2 − 3y2 and 2x2 − y2+ 9.

Answer:

Sum of x2 - 3y2 and 2x2 - y2 + 9
= (x2 - 3y2) + (2x2 - y2 + 9)
= x2 + 2x2 - 3y2 - y2+ 9
= 3x2 - 4y2 + 9

Now, required expression = (x2 + 2xy + y2) + (3x2 - 4y2 + 9)
                                       = x2 + 3x2 + 2xy + y2 - 4y2 + 9
                                       = 4x2 + 2xy  - 3y2 + 9

Page No 7.14:

Question 9:

Add a3 + b3 − 3 to the sum of 2a3 − 3b3 − 3ab + 7 and −a3 + b3 + 3ab −9.

Answer:

First, we need to find the sum of 2a3 - 3b3 - 3ab + 7 and - a3 + b3 + 3ab - 9.
= (2a3 - 3b3 - 3ab + 7) + (- a3 + b3 + 3ab - 9)
Collecting positive and negative like terms together, we get
= 2a3 - a3- 3b3 + b3 - 3ab + 3ab + 7 - 9
= a3 - 2b3 - 2

Now, the required expression = (a3 + b3 - 3) + (a3 - 2b3 - 2).
                                  = a3 + a3 + b3- 2b3 - 3 - 2
                                  = 2a3 - b3 - 5

Page No 7.14:

Question 10:

Subtract:
(i) 7a2b from 3a2b
(ii) 4xy from −3xy

Answer:

(i) Required expression  =  3a2b - 7a2b
                                        = (3 - 7)a2b
                                        = - 4a2b

(ii) Required expression  = - 3 xy - 4xy
                                         = - 7xy
                                      

Page No 7.14:

Question 11:

Subtract:
(i) −4x from 3y
(ii) −2x from −5y

Answer:

(i) Required expression = (3y) - (-4x)
                                       = 3y + 4x

(ii) Required expression = (-5y) - (-2x)
                                        = -5y + 2x

Page No 7.14:

Question 12:

Subtract:
(i) 6x3-7x2+5x-3 from 4-5x+6x2-8x3
(ii) -x2-3z from 5x2-y+z+7
(iii) x3+2x2y+6xy2-y3 from y3-3xy2-4x2y

Answer:

(i) Required expression = (4 - 5x + 6x2 - 8x3) - (6x3 - 7x2 + 5x - 3)
                                      = 4 - 5x + 6x2 - 8x3 - 6x3 + 7x2 - 5x + 3
                                      = - 8x3- 6x3 + 7x2 + 6x2- 5x - 5x + 3 + 4
                                      = - 14x3 + 13x2 - 10x +7

(ii) Required expression  = (5x2 - y + z + 7) - (- x2 - 3z)
                                        = 5x2 - y + z + 7 + x2 + 3z
                                        = 5x2+ x2 - y + z + 3z + 7
                                        = 6x2 - y + 4z + 7

(iii) Required expression = (y3 - 3xy2 - 4x2y) - (x3 + 2x2y + 6xy2 - y3)
                                         = y3 - 3xy2 - 4x2y - x3 - 2x2y - 6xy2 + y3
                                         = y3 + y3 - 3xy2- 6xy2 - 4x2y - 2x2y - x3
                                         = 2y3- 9xy2 - 6x2y - x3

Page No 7.14:

Question 13:

From
(i) p3 − 4 + 3p2, take away 5p2 − 3p3 + p − 6
(ii) 7 + xx2, take away 9 + x + 3x2 + 7x3
(iii) 1 − 5y2, take away y3 + 7y2 + y + 1
(iv) x3 − 5x2 + 3x + 1, take away 6x2 − 4x3 + 5 + 3x

Answer:

 (i) Required expression = (p3 - 4 + 3p2) - (5p2 - 3p3 + p - 6)
                                        = p3 - 4 + 3p2 - 5p2 + 3p3 - p + 6
                                        = p3 + 3p3 + 3p2 - 5p2- p - 4+ 6
                                        = 4p3 - 2p2 - p + 2

(ii) Required expression = (7 + x - x2) - (9 + x + 3x2 + 7x3)
                                        = 7 + x - x2 - 9 - x - 3x2 - 7x3
                                        = - 7x3- x2 - 3x2 + 7 - 9
                                        = - 7x3 - 4x2 - 2

(iii) Required expression = (1 - 5y2) - (y3+ 7y2 + y + 1)
                                         = 1 - 5y2 - y3 - 7y2 - y - 1
                                         = - y3- 5y2 - 7y2 - y
                                         = - y3- 12y2 - y

(iv) Required expression = (x3 - 5x2 + 3x + 1) - (6x2 - 4x3 + 5 +3x)
                                         = x3 - 5x2 + 3x + 1 - 6x2 + 4x3 - 5 - 3x
                                         = x3 + 4x3 - 5x2 - 6x2 + 1 - 5
                                         = 5x3 - 11x2 - 4

Page No 7.14:

Question 14:

From the sum of 3x2 − 5x + 2 and −5x2 − 8x + 9 subtract 4x2 − 7x + 9.

Answer:

Required expression = {(3x2 - 5x + 2) + (- 5x2 - 8x + 9)} - (4x2 - 7x + 9)
                                  = {3x2 - 5x + 2 - 5x2 - 8x + 9} -  (4x2 - 7x + 9)
                                  = {3x2 - 5x2 - 5x - 8x + 2 + 9} -  (4x2 - 7x + 9)
                                  = {- 2x2 - 13x +11} - (4x2 - 7x + 9)
                                  = - 2x2 - 13x + 11 - 4x2 + 7x - 9
                                  = - 2x2 - 4x2 - 13x + 7x + 11 - 9
                                  = - 6x2 - 6x + 2

Page No 7.14:

Question 15:

Subtract the sum of 13x − 4y + 7z and −6z + 6x + 3y from the sum of 6x − 4y − 4z and 2x + 4y − 7.

Answer:

Sum of (13x - 4y + 7z) and ( - 6z + 6x + 3y)
= {(13x - 4y + 7z) + (- 6z + 6x + 3y)
={13x - 4y + 7z - 6z + 6x + 3y}
= {13x + 6x - 4y + 3y + 7z - 6z}
= 19x - y + z

Sum of (6x − 4y − 4z) and (2x + 4y − 7)
= (6x − 4y − 4z) + (2x + 4y − 7)
= 6x − 4y − 4z + 2x + 4y − 7
= 8x - 4z - 7

Now, required expression = {(8x - 4z - 7) - (19x - y + z)}
                                  = 8x - 4z - 7 - 19x + y - z
                                  = 8x - 19x + y - 4z - z - 7
                                  = - 11x + y - 5z - 7

Page No 7.14:

Question 16:

From the sum of x2 + 3y2 − 6xy, 2x2y2 + 8xy, y2 + 8 and x2 − 3xy subtract −3x2 + 4y2xy + xy + 3.

Answer:

Sum of (x2 + 3y2 - 6xy), (2x2 - y2 + 8xy), (y2 + 8) and (x2 - 3xy)
={(x2 + 3y2 - 6xy) + (2x2 - y2 + 8xy) + ( y2 + 8) + (x2 - 3xy)}
={x2 + 3y2 - 6xy + 2x2 - y2 + 8xy + y2 + 8 + x2 - 3xy}
= {x2+ 2x2+ x2 + 3y2- y2 + y2- 6xy + 8xy - 3xy + 8}
= 4x2 + 3y2 - xy + 8

Now, required expression = (4x2 + 3y2 - xy + 8) - (- 3x2 + 4y2 - xy + x - y + 3)
                                  = 4x2 + 3y2 - xy + 8 + 3x2 - 4y2 + xy - x + y - 3
                                  = 4x2 + 3x2+ 3y2- 4y2- x + y - 3 + 8
                                  = 7x2 - y2- x + y + 5

Page No 7.14:

Question 17:

What should be added to xy − 3yz + 4zx to get 4xy − 3zx + 4yz + 7?

Answer:

The required expression can be got by subtracting xy - 3yz + 4zx from 4xy - 3zx + 4yz + 7.
Therefore, required expression = (4xy - 3zx + 4yz + 7) - (xy - 3yz + 4zx)
                                  = 4xy - 3zx + 4yz + 7 - xy + 3yz - 4zx
                                  = 4xy - xy - 3zx - 4zx + 4yz + 3yz + 7
                                  = 3xy - 7zx + 7yz + 7

Page No 7.14:

Question 18:

What should be subtracted from x2 xy + y2x + y + 3 to obtain −x2 + 3y2 − 4xy + 1?

Answer:

Let 'M' be the required expression. Then, we have
x2 - xy + y2 - x + y + 3 - M = - x2 + 3y2 - 4xy + 1
Therefore,
M = (x2 - xy + y2 - x + y + 3) - (- x2 + 3y2 - 4xy + 1)
    = x2 - xy + y2 - x + y + 3 + x2 - 3y2 + 4xy - 1
Collecting positive and negative like terms together, we get
x2 + x2- xy + 4xy + y2- 3y2 - x + y + 3 - 1
= 2x2 + 3xy- 2y2- x + y + 2

Page No 7.14:

Question 19:

How much is x − 2y + 3z greater than 3x + 5y − 7?

Answer:

Required expression  = (x - 2y + 3z) - (3x + 5y - 7)
                                   =  x - 2y + 3z - 3x - 5y + 7
Collecting positive and negative like terms together, we get
x - 3x - 2y - 5y + 3z + 7
= - 2x - 7y + 3z + 7

Page No 7.14:

Question 20:

How much is x2 − 2xy + 3y2 less than 2x2 − 3y2 + xy?

Answer:

Required expression = (2x2 - 3y2 + xy) - (x2 - 2xy + 3y2)
                                  = 2x2 - 3y2 + xy - x2 + 2xy - 3y2
Collecting positive and negative like terms together, we get
2x2 - x2 - 3y2 - 3y2 + xy + 2xy
 = x2 - 6y2 + 3xy

Page No 7.14:

Question 21:

How much does a2 − 3ab + 2b2 exceed 2a2 − 7ab + 9b2?

Answer:

Required expression = (a2 - 3ab + 2b2) - (2a2 - 7ab + 9b2)
                                  = a2 - 3ab + 2b2 - 2a2 + 7ab - 9b2
Collecting positive and negative like terms together, we get
                                  =  a2 - 2a2  - 3ab + 7ab  + 2b2 -  9b2   
                                  = - a2 + 4ab - 7b2

Page No 7.14:

Question 22:

What must be added to 12x3 − 4x2 + 3x − 7 to make the sum x3 + 2x2 − 3x + 2?

Answer:

Let 'M' be the required expression. Thus, we have
12x3 - 4x2 + 3x - 7 + M = x3 + 2x2 - 3x + 2
Therefore,
M = (x3 + 2x2 - 3x + 2) - (12x3 - 4x2 + 3x - 7)
    =  x3 + 2x2 - 3x + 2 - 12x3 + 4x2 - 3x + 7
Collecting positive and negative like terms together, we get
x3- 12x3 + 2x2 + 4x2 - 3x - 3x + 2 + 7
= - 11x3 + 6x2 - 6x + 9

Page No 7.14:

Question 23:

If P = 7x2 + 5xy − 9y2, Q = 4y2 − 3x2 − 6xy and R = −4x2 + xy + 5y2, show that P + Q + R = 0.

Answer:

We have
P + Q + R = (7x2 + 5xy - 9y2) + (4y2 - 3x2 - 6xy) + (- 4x2 + xy + 5y2)
                 = 7x2 + 5xy - 9y2 + 4y2 - 3x2 - 6xy - 4x2 + xy + 5y2
Collecting positive and negative like terms together, we get
7x2- 3x2 - 4x2 + 5xy - 6xy + xy - 9y2 + 4y2 + 5y2
= 7x2- 7x+ 6xy - 6xy  - 9y2 + 9y2
= 0

Page No 7.14:

Question 24:

If P = a2 b2 + 2ab, Q = a2 + 4b2 − 6ab, R = b2 + b, S = a2 − 4ab and T = −2a2 + b2ab + a. Find P + Q + R + S − T.

Answer:

 We have
P + Q + R + S - T = {(a2 - b2 + 2ab) + (a2 + 4b2 - 6ab) + (b2 + b) + (a2 - 4ab)} - (-2a2 + b2 - ab + a)
                             = {a2 - b2 + 2ab + a2 + 4b2 - 6ab + b2 + b + a2 - 4ab}- (- 2a2 + b2 - ab + a)
                             = {3a2 + 4b2 - 8ab + b } - (-2a2 + b2 - ab + a)
                             = 3a2+ 4b2 - 8ab + b + 2a2 - b2 + ab - a
Collecting positive and negative like terms together, we get
3a2 + 2a2 + 4b2 - b2 - 8ab + ab - a + b
= 5a2 + 3b2- 7ab - a + b



Page No 7.16:

Question 1:

Place the last two terms of the following expressions in parentheses preceded by a minus sign:
(i) x + y − 3z + y
(ii) 3x 2y − 5z − 4
(iii) 3a 2b + 4c − 5
(iv) 7a + 3b + 2c + 4
(v) 2a2 b2 − 3ab + 6
(vi) a2 + b2 c2 + ab − 3ac

Answer:

We have
(i) x + y − 3z + y = x  + y − (3z - y )
(ii) 3x − 2y − 5z − 4 = 3x - 2y - (5z + 4)
(iii) 3a − 2b + 4c − 5 = 3a - 2b - (- 4c + 5)
(iv) 7a + 3b + 2c + 4 = 7a + 3b - (- 2c - 4)
(v) 2a2 − b2 − 3ab + 6 = 2a2 − b2 − (3ab - 6)
(vi) a2 + b2 − c2 + ab − 3ac = a2 + b2 − c2 - (- ab + 3ac)

Page No 7.16:

Question 2:

Write each of the following statements by using appropriate grouping symbols:
(i) The sum of a b and 3a − 2b + 5 is subtracted from 4a + 2b − 7.
(ii) Three times the sum of 2x + y − {5 − (x − 3y)} and 7x − 4y + 3 is subtracted from 3x − 4y + 7.
(iii) The subtraction of x2y2 + 4xy from 2x2 + y2 − 3xy is added to 9x2 − 3y2xy.

Answer:

(i) The sum of a − b and 3a − 2b + 5 = {(a - b) + (3a − 2b + 5)}.
     This is subtracted from 4a + 2b - 7.
     Thus, the required expression is {4a + 2b - 7) - {(a - b) + (3a − 2b + 5)}.

(ii) Three times the sum of 2x + y − {5 − (x − 3y)} and 7x − 4y + 3 = 3[(2x + y) − {5 − (x − 3y)} + (7x − 4y + 3)].
      This is subtracted from 3x - 4y +7.
      Thus, the required expression is (3x - 4y +7) - 3[(2x + y) − {5 − (x − 3y)} + (7x − 4y + 3)].

(iii) The product of subtraction of x2 − y2 + 4xy from 2x2 + y2 − 3xy is given by {(2x2 + y2 − 3xy) - (x2 − y2 + 4xy)}.
       When the above equation is added to 9x2 − 3y2 − xy, we get
       {(2x2 + y2 − 3xy) - (x2 − y2 + 4xy)} + (9x2 − 3y2 − xy)



Page No 7.20:

Question 1:

Simplify each of the following algebraic expressions by removing grouping symbols.
2x + (5x − 3y)

Answer:

We have 
   2x + (5x − 3y)
Since the '+' sign precedes the parentheses, we have to retain the sign of each term in the parentheses when we remove them.
= 2x + 5x - 3y
= 7x - 3y

Page No 7.20:

Question 2:

Simplify each of the following algebraic expressions by removing grouping symbols.
3x − (y − 2x)

Answer:

We have
3x − (y − 2x)
Since the '-' sign precedes the parentheses, we have to change the sign of each term in the parentheses when we remove them. Therefore, we have
3x − y + 2x
= 5x - y

Page No 7.20:

Question 3:

Simplify each of the following algebraic expressions by removing grouping symbols.
5a − (3b − 2a + 4c)

Answer:

We have
5a − (3b − 2a + 4c)
Since the '-' sign precedes the parentheses, we have to change the sign of each term in the parentheses when we remove them.
= 5a - 3b + 2a - 4c
= 7a - 3b - 4c

Page No 7.20:

Question 4:

Simplify each of the following algebraic expressions by removing grouping symbols.
−2 (x2y2 + xy) − 3(x2 + y2xy)

Answer:

We have
− 2(x2 − y2 + xy) − 3(x2 + y2 − xy)
Since the '-' sign precedes the parentheses, we have to change the sign of each term in the parentheses when we remove them.
= - 2x2 + 2y2 - 2xy - 3x2 - 3y2 + 3xy
= - 2x2 - 3x2 + 2y2- 3y2 - 2xy + 3xy
= - 5x2 - y2 + xy

Page No 7.20:

Question 5:

Simplify each of the following algebraic expressions by removing grouping symbols.
3x + 2y − {x − (2y − 3)}

Answer:

We have
3x + 2y − {x − (2y − 3)}
First, we have to remove the small brackets (or parentheses): ( ). Then, we have to remove the curly brackets (or braces): { }.
Therefore,
= 3x + 2y − {x − 2y + 3}
= 3x + 2y − x + 2y - 3
= 2x + 4y - 3

Page No 7.20:

Question 6:

Simplify each of the following algebraic expressions by removing grouping symbols.
5a − {3a − (2 − a) + 4}

Answer:

We have
5a − {3a − (2 − a) + 4}
First, we have to remove the small brackets (or parentheses): ( ). Then, we have to remove the curly brackets (or braces): { }.
Therefore,
= 5a − {3a − 2 + a + 4}
= 5a − 3a + 2 - a - 4
= 5a - 4a - 2
= a - 2

Page No 7.20:

Question 7:

Simplify each of the following algebraic expressions by removing grouping symbols.
a − [b − {a − (b − 1) + 3a}]

Answer:

First we have to remove the parentheses, or small brackets, ( ), then the curly brackets, { }, and then the square brackets [ ].
Therefore, we have
a - [b - {a - (b - 1) + 3a}]
= a - [b - {a - b + 1 + 3a}]
= a - [b - {4a - b + 1}]
= a - [b - 4a + b - 1]
= a - [2b - 4a - 1]
= a - 2b + 4a + 1
= 5a - 2b + 1

Page No 7.20:

Question 8:

Simplify each of the following algebraic expressions by removing grouping symbols.
a − [2b − {3a − (2b − 3c)}]

Answer:

First we have to remove the small brackets, or parentheses, ( ), then the curly brackets, { }, and then the square brackets, [ ].
Therefore, we have
a - [2b - {3a - (2b - 3c)}]
= a - [2b - {3a - 2b + 3c}]
= a - [2b - 3a + 2b - 3c]
= a - [4b - 3a - 3c]
= a - 4b + 3a + 3c
= 4a - 4b + 3c

Page No 7.20:

Question 9:

Simplify each of the following algebraic expressions by removing grouping symbols.
x + [5y − {2x − (3y − 5x)}]

Answer:

First we have to remove the small brackets, or parentheses, ( ), then the curly brackets, { }, and then the square brackets, [ ].
Therefore, we have
- x + [5y - {2x - (3y - 5x)}]
= - x + [5y - {2x - 3y + 5x}]
= - x + [5y - {7x - 3y}]
= - x + [5y - 7x + 3y]
= - x + [8y - 7x]
= - x + 8y - 7x
= - 8x + 8y

Page No 7.20:

Question 10:

Simplify each of the following algebraic expressions by removing grouping symbols.
2a − [4b − {4a − 3(2ab)}]

Answer:

First we have to remove the small brackets, or parentheses, ( ), then the curly brackets, { }, and then the square brackets, [ ].
Therefore, we have
2a - [4b - {4a - 3(2a - b)}]
= 2a - [4b - {4a - 6a + 3b}]
= 2a - [4b - {- 2a + 3b}]
= 2a - [4b + 2a - 3b]
= 2a - [b + 2a]
= 2a - b - 2a
= - b

Page No 7.20:

Question 11:

Simplify each of the following algebraic expressions by removing grouping symbols.
a − [a + {a + b − 2a − (a − 2b)} − b]

Answer:

First we have to remove the small brackets, or parentheses, ( ), then the curly brackets,{ }, and then the square brackets, [ ].
Therefore, we have
- a - [a + {a + b - 2a - (a - 2b)} - b]
= - a - [a + {a + b - 2a - a + 2b} - b]
= - a - [a + {- 2a + 3b} - b]
= - a - [a - 2a + 3b - b]
= - a - [- a + 2b]
= - a + a - 2b
= - 2b

Page No 7.20:

Question 12:

Simplify each of the following algebraic expressions by removing grouping symbols.
2x − 3y − [3x − 2y − {x − z − (x − 2y)}]

Answer:

First we have to remove the small brackets, or parentheses, ( ), then the curly brackets, { }, and then the square brackets, [ ].
Therefore, we have
2x - 3y - [3x - 2y - {x - z - (x - 2y)}]
= 2x - 3y - [3x - 2y - {x - z - x + 2y}]
= 2x - 3y - [3x - 2y - {- z + 2y}]
= 2x - 3y - [3x - 2y + z - 2y]
= 2x - 3y - [3x - 4y + z]
= 2x - 3y - 3x + 4y - z
= - x + y - z

Page No 7.20:

Question 13:

Simplify each of the following algebraic expressions by removing grouping symbols.
5 + [x − {2y − (6x + y − 4) + 2x} − {x − (y − 2)}]

Answer:

First we have to remove the small brackets, or parentheses, ( ), then the curly brackets, { }, and then the square brackets, [ ].
Therefore, we have
5 + [x - {2y - (6x + y - 4) + 2x} - {x - (y - 2)}]
= 5 + [x - {2y - 6x - y + 4 + 2x} - {x - y + 2}]
= 5 + [x - {y - 4x + 4} - {x - y + 2}]
= 5 + [x - y + 4x - 4 - x + y - 2]
= 5 + [4x - 6]
= 5 + 4x - 6
= 4x - 1

Page No 7.20:

Question 14:

Simplify each of the following algebraic expressions by removing grouping symbols.
x2 − [3x + {2x − (x2 − 1) + 2}]

Answer:

First we have to remove the small brackets, or parentheses, ( ), then the curly brackets, { }, and then the square brackets, [ ].
Therefore, we have
x2 - [3x + {2x - (x2 - 1)} + 2]
= x2 - [3x + {2x - x2 + 1} + 2]
= x2 - [3x + 2x - x2 + 1+ 2]
= x2 - [5x - x2 + 3]
= x2 - 5x + x2 - 3
= 2x2 - 5x - 3

Page No 7.20:

Question 15:

Simplify each of the following algebraic expressions by removing grouping symbols.
20 − [5xy + 3{x2 − (xyy) − (xy)}]

Answer:

First we have to remove the small brackets, or parentheses, ( ), then the curly brackets, { }, and then the square brackets, [ ].
Therefore, we have
20 - [5xy + 3{x2 - (xy - y) - (x - y)}]
= 20 - [5xy + 3{x2 - xy + y - x + y}]
= 20 - [5xy + 3{x2 - xy + 2y - x}]
= 20 - [5xy + 3x2 - 3xy + 6y - 3x]
= 20 - [2xy + 3x2 + 6y - 3x]
= 20 - 2xy - 3x2 - 6y + 3x
= - 3x2 - 2xy - 6y + 3x + 20

Page No 7.20:

Question 16:

Simplify each of the following algebraic expressions by removing grouping symbols.
85 − [12x − 7(8x − 3) − 2 {10x − 5(2 − 4x)}]

Answer:

First we have to remove the small brackets, or parentheses, ( ), then the curly brackets, { }, and then the square brackets, [ ].
Therefore, we have
85 - [12x - 7(8x - 3) - 2{10x - 5(2 - 4x)}]
= 85 - [12x - 56x + 21 - 2{10x - 10 + 20x}]
= 85 - [12x - 56x + 21 - 2{30x - 10}]
= 85 - [12x - 56x + 21 - 60x + 20]
= 85 - [12x - 116x + 41]
= 85 - [- 104x + 41]
= 85 + 104x - 41
= 44 + 104x

Page No 7.20:

Question 17:

Simplify each of the following algebraic expressions by removing grouping symbols.
xy [yzzx − {yx − (3yxz) − (xyzy)}]

Answer:

First we have to remove the small brackets, or parentheses, ( ), then the curly brackets, { }, and then the square brackets, [ ].
Therefore, we have
xy - [yz - zx - {yx - (3y - xz) - (xy - zy)}]
= xy - [yz - zx - {yx - 3y + xz - xy + zy}]
= xy - [yz - zx - {- 3y + xz + zy}]
= xy - [yz - zx + 3y - xz - zy]
= xy - [- zx + 3y - xz]
= xy - [- 2zx + 3y]
= xy + 2xz - 3y



Page No 7.7:

Question 1:

Identify the monomials, binomials, trinomials and quadrinomials from the following expressions:
(i) a2
(ii) a2 b2
(iii) x3 + y3 + z3
(iv) x3 + y3 + z3 + 3xyz
(v) 7 + 5
(vi) abc + 1
(vii) 3x − 2 + 5
(viii) 2x − 3x + 4
(ix) xy + yz + zx
(x) ax3 + bc2 + cx + d

Answer:

The monomials, binomials, trinomials and quadrinomials are as follows.
(i) a2 is a monomial expression as it contains one term only.
(ii) a2 - b2 is a binomial expression as it contains two terms.
(iii) x3 + y3 + z3 is a trinomial expression as it contains three terms.
(iv) x3 + y3 + z3 + xyz is a quadrinomial expression as it contains four terms.
(v) 7 + 5 = 12 is a monomial expression as it contains one term only.
(vi) abc +1 is a binomial expression as it contains two terms.
(vii) 3x - 2 + 5 = 3x + 3 is a binomial expression as it contains two terms.
(viii) 2x - 3y + 4 is a trinomial expression as it contains three terms.
(ix) xy + yz + zx is a trinomial expression as it contains three terms.
(x) ax3 +bx2 +cx + d is a quadrinomial expression as it contains four terms.



Page No 7.8:

Question 2:

Write all the terms of each of the following algebraic expressions:
(i) 3x
(ii) 2x 3
(iii) 2x2 − 7
(iv) 2x2 + y2 − 3xy + 4

Answer:

The terms of each of the given algebraic expressions are as follows.
(i) 3x is the only term of the given algebraic expression.
(ii) 2x and -3 are the terms of the given algebraic expression.
(iii) 2x2 and -7 are the terms of the given algebraic expression.
(iv) 2x2, y2, -3xy  and 4 are the terms of the given algebraic expression.

Page No 7.8:

Question 3:

Identify the like terms and also mention the numerical coefficients of those terms:
(i) 4xy, 5x2y, −3yx, 2xy2
(ii) 7a2bc, -3ca2b, -52abc2, 32abc2, -43cba2

Answer:

       Like terms                                          Numerical coefficients
(i)   4xy, -3yx                                                4, -3
(ii) {7a2bc ,-3ca2b, -43cba2}                      { 7, -3,-43}
      {-52abc2,32abc2 }                                 {-52,32}

Page No 7.8:

Question 4:

Identify the like terms in the following algebraic expressions:
(i) a2+b2-2a2+c2+4a
(ii) 3x+4xy-2yz+52zy
(iii) abc+ab2c+2acb2+3c2ab+b2ac-2a2bc+3cab2

Answer:

The like terms in the given algebraic expressions are as follows.
(i) The like terms in the given algebraic expression are a2  and -2a2.

(ii) The like terms in the given algebraic expression are -2yz and 52zy.
(iii) The like terms in the given algebraic expression are ab2c, 2acb2, b2ac and 3cab2.

Page No 7.8:

Question 5:

Write the coefficient of x in the following:
(i) −12x
(ii) −7xy
(iii) xyz
(iv) −7ax

Answer:

 The coefficients of x are as follows.
(i) The numerical coefficient of x is -12.
(ii) The numerical coefficient of x is -7y.
(iii) The numerical coefficient of x is yz.
(iv) The numerical coefficient of x is -7a.

Page No 7.8:

Question 6:

Write the coefficient of x2 in the following:
(i) −3x2
(ii) 5x2yx
(iii) 57x2z
(iv) -32ax2+yx

Answer:

The coefficients of x2 are as follows.
 (i) The numerical coefficient of x2 is -3.
(ii) The numerical coefficient of x2 is 5yz.

(iii) The numerical coefficient of x2 is 57z.
(iv) The numerical coefficient of x2 is -32a.

Page No 7.8:

Question 7:

Write the coefficient of:
(i) y in −3y
(ii) a in 2ab
(iii) z in −7xyz
(iv) p in −3pqr
(v) y2 in 9xy2z
(vi) x3 in x3 + 1
(vii) x2 in −x2

Answer:

 The coefficients are as follows.

 (i) The coefficient of  y is -3.
 (ii) The coefficient of  a is 2b.
 (iii) The coefficient of  z is -7xy.
 (iv) The coefficient of  p is -3qr.
 (v) The coefficient of  y2 is 9xz.
 (vi) The coefficient of  x3 is 1.
 (vii) The coefficient of  -x2 is - 1.

Page No 7.8:

Question 8:

Write the numerical coefficient of each of the following:
(i) xy
(ii) −6yz
(iii) 7abc
(iv) −2x3y2z

Answer:

The numerical coefficient of each of the given terms is as follows.
(i) The numerical coefficient in the term xy is 1.
(ii) The numerical coefficient in the term -6yz is - 6.
(iii) The numerical coefficient in the term 7abc is 7.
(iv) The numerical coefficient in the term -2x3y2z is - 2.

Page No 7.8:

Question 9:

Write the numerical coefficient of each term in the following algebraic expressions:
(i) 4x2y-32xy+52xy2
(ii) -53x2y+74xyz+3

Answer:

The numerical coefficient of each term in the given algebraic expressions is as follows.
             Term                         Coefficient

(i)       4x2y                                   4
                          
         -32xy                              -32
           52xy2                                 52

(ii)    -53x2y                            -53
          74xyz                                 74
           3                                        3
        

Page No 7.8:

Question 10:

Write the constant term of each of the following algebraic expressions:
(i) x2y xy2 + 7xy − 3
(ii) a3 − 3a2 + 7a + 5

Answer:

The constant term of each of the given algebraic expressions is as follows.
(i) The constant term in the given algebraic expression is -3.
(ii) The constant term in the given algebraic expression is 5.

Page No 7.8:

Question 11:

Evaluate each of the following expressions for x = − 2, y = −1, z = 3:
(i) xy+yz+zx
(ii) x2+y2+z2-xy-yz-zx

Answer:

We have x = −2, y = −1 and z = 3.
Thus,
(i)
     xy+yz+zx=-2-1+-13+3-2=2-13-32=12-2-96=12-116=16

(ii) x2 + y2 + z2- xy - yz - zx
     = (-2)2 + (-1)2 + (3)2 - (-2)(-1) - (-1)(3) - (3)(-2)
     = 4 + 1 + 9 - 2 + 3 + 6
     = (4 + 1 + 9 + 3 + 6) - 2
     = 23 - 2 = 21

Page No 7.8:

Question 12:

Evaluate each of the following algebraic expressions for x = 1, y = −1, z = 2, a = −2, b = 1, c = −2:
(i) ax + by + cz
(ii) ax2 + by2cz2
(iii) axy + byz + cxy

Answer:

We have x = 1, y = −1, z = 2, a = −2, b = 1 and c = −2.
Thus,
(i) ax + by + cz
    = (-2)(1) + (1)(-1) + (-2)(2)
    = - 2 + (- 1) + (- 4)
    = - 2 -1 - 4 = -7

(ii) ax2 + by2 - cz2
    = (-2)(1)2 + (1)(-1)2 - (-2)(2)2
    = -2 x 1 + 1 - (-2 x 4)
    = -2 + 1 - (-8)
    = -2 + 1 + 8
    = -2 + 9
    = 7

(iii) axy + byz + cxy
     = (-2)(1)(-1) + (1)(-1)(2) + (-2)(1)(-1)
     = 2 + (-2) + 2
     = 2 - 2 + 2
     = 4 - 2
     = 2



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