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Pair of Linear Equations in Two Variables

We know that many day to day situations can be expressed in the form of equations. However, the importance of these equations can only be understood, if we are able to solve them.

Let us consider an example.

We can also represent the solution of an equation graphically. Let us see how.

First of all, we need to find the solution of the given equation. Then, we need to represent it on a number line.

Let us start working with the equation 2x − 3 = −11.

Here, we have

2x − 3 = −11

⇒ 2x − 3 + 3 = −11 + 3(Adding 3 to both sides)

⇒ 2x = −8

(Dividing both sides by 2)

x = −4

The solution is represented by a thick dot on the number line as shown below.

Let us now look at some more examples to understand the concept better.

Example 1:

Solve the following equations.

  1. 9x + 3 = 21
  2. 3y – 4 = 14
  3. –10 = 23m

Solution:

1. 9x + 3 = 21

On subtracting 3 from both sides, we obtain

9x + 3 – 3 = 21 – 3

⇒ 9x = 18

On dividing both sides by 9, we obtain

⇒ 9x ÷ 9 = 18 ÷ 9

x = 2

2. 3y – 4 = 14

On adding 4 to both sides, we obtain

3y – 4 + 4 = 14 + 4

⇒ 3y = 18

On dividing both sides by 3, we obtain

3y ÷ 3 = 18 ÷ 3

y = 6

3. –10 =

On multiplying both sides by 3, we obtain

–10 × 3 =

⇒ –30 = 2m

On dividing both sides by 2, we obtain

–30 ÷ 2 = 2m ÷ 2

⇒ –15 = m

m = –15 

Example 2:

Solve the following equations. Also, represent their solutions graphically.

  1. 3p + 2 = 35
  2. z=

Solution:

  1. 3p + 2 = 35

On subtracting 2 from both the sides, we obtain

3p + 2 − 2 = 35 − 2

⇒ 3p = 33

On dividing both the sides by 3, we obtain

p = 11

The solution is represented by a thick dot on the number line as shown below.

  1. z=

On adding to both the sides, we obtain

z =

z =

z = 9

The solution is represented by a thick dot on the number line as shown below.

Example 3:

The average of three numbers is 15. If two of these numbers are 12 and 24, then find the third number.

Solution:

Let the required number be x.

It is given that the average of 12, 24, and x is 15. Therefore,

On multiplying both the sides with 3, we obtain

x + 36 = 45

On subtracting 36 from both the sides, we obtain

x + 36 − 36 = 45 − 36

x = 9

Thus, the required number is 9.

Example 4:

Mohit and Rohit are friends. Mohit has Rs 5 more than 3 times the money Rohit has. If Mohit has Rs 23, then how much money does Rohit have?

Solution:

Let us assume that Rohit has Rs x.

Three times the money with Rohit = Rs 3x

According to the question, Mohit has Rs (3x + 5).

It is given that Mohit has Rs 23.

3x + 5 = 23

On subtracting 5 from both the sides, we obtain

3x + 5 − 5 = 23 − 5

3x = 18

Now, on dividing both the sides by 3, we obtain

x = 6

Thus, Rohit has Rs 6.

We know how to solve an equation by performing the same mathematical operation on both its sides. There is one more method a…

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