In this chapter we will learn to solve linear equation with two variables using substitution method.

To understand the chapter, you should have fair knowledge of linear and simultaneous equations.

## Substitution Method

This method works when we have** two linear equations with two variables**.

Using the method **we can find common point that satisfies both the given equation**.

Let we have two linear equations with x & y variables.

The equation can be solved by **following below steps**;

(a) **Find value of x with respect to y or vice – versa** using one of the given equation.

(b) **Now put this value of x in the second equa**tion.

On solving the equation, you will get exact value of y.

(c)** Now put this value of y in any of the given equation and you will get exact value of x.**

By following these three steps, you can solve given set of linear equations.

Let us see some solved examples for better clarity.

**Example 01**

Solve the below equations using substitution method.

5x + 2y = 16

x + 8y = 26

**Solution****Take the second equation** and **find the value of x with respect to y**.

x + 8y = 26

x = 26 – 8y**Now substitute this value of x in first equation**.

5x + 2y = 16

5 (26 – 8y) + 2y = 16

130 – 40y + 2y = 16

-38y = 16- 130

-38y = -114

y = 114 / 38**y = 3**

Here we got the exact value of y.

**Now put this y value in any of the given equation & you will get value of x**.

Taking first equation.

5x + 2y = 16

5x + 2(3) = 16

5x + 6 = 16

5x = 16 – 6

5x = 10

x = 10 / 5**x = 2**

Hence, **(2, 3) is the solution of given set of equations**.

**Example 02**

Solve the below equations using substitution method.

2x + y = 1

4x – 2y = 6

**Solution****Taking first equation** and find **value of y with respect of x**.

2x + y = 1

y = 1 – 2x

**Now substitute this y value in second equation**.

4x – 2y = 6

4x – 2 (1 – 2x) = 6

4x – 2 + 4x = 6

8x = 6 + 2

8x = 8

**x = 1**

Here we have got the exact value of x.

Put the value of x in any of the given equation to find exact value of y.

**Taking first equation**

2x + y = 1

2(1) + y = 1

2 + y = 1

y = 1 – 2**y = -1**

Hence, **(1, -1) is the solution of given equations.**

**Example 03**

Solve the below equations using substitution method.

3x – 2y = 4

-2x + 2y = -1

**Solution****Take first equation and find value of x with respect of y**.

3x – 2y = 4

3x = 4 + 2y

x = \mathtt{\frac{4+2y}{3}}

**Now substitute this value of x in second equation.**

-2x + 2y = -1

\mathtt{-2\left(\frac{4+2y}{3}\right) +2y=-1}\\\ \\ \mathtt{\frac{-8-4y}{3} +2y=-1}\\\ \\ \mathtt{\frac{-8-4y+6y}{3} =-1}\\\ \\ \mathtt{-8+2y=-3}\\\ \\ \mathtt{2y=\ -3+8}\\\ \\ \mathtt{y\ =\ \frac{5}{2}}

Here we have got exact value of y. Now put this value in any of the given equation to find exact value of x.

Taking first equation.

\mathtt{3x\ -\ 2y\ =\ 4}\\\ \\ \mathtt{3x\ -\ 2\left(\frac{5}{2}\right) =4}\\\ \\ \mathtt{3x\ -5\ =\ 4}\\\ \\ \mathtt{3x\ =\ 4\ +5}\\\ \\ \mathtt{3x\ =\ 9}\\\ \\ \mathtt{x\ =\ \frac{9}{3}}\\\ \\ \mathtt{x\ =\ 3}

Hence, ( 3, 5/2) is the solution.

**Example 04**

Solve the below equations using substitute method.

x – y = 2

y + 2x = 4

**Solution****Take the first equation and find the value of x**.

x – y = 2

x = y + 2

**Now substitute this value of x in second equation.**

y + 2x = 4

y + 2 (y + 2) = 4

y + 2y + 4 = 4

3y = 4 – 4

3y = 0**y = 0**

We have got exact value of y. Put this value of y in any of the given equation to find value of x.**Taking first equation**.

x – y = 2

x – 0 = 2**x = 2**

Hence, **(2, 0) is the solution of given expression.**

**Example 05**

Solve the below equation using substitute method.

2x – 6y = -6

7x – 8y = 5

**Solution****Take the first equation and find value of y with respect to x**.

\mathtt{2x\ -\ 6y\ =\ -6}\\\ \\ \mathtt{2x\ =\ 6y\ -\ 6}\\\ \\ \mathtt{x\ =\ \frac{6y-6}{2}}\\\ \\ \mathtt{x\ =\ \frac{2( 3y-3)}{2}}\\\ \\ \mathtt{x\ =\ 3y-3}

**Put this value of x in second equation.**

7x – 8y = 5

7 (3y – 3) – 8y = 5

21y -21 – 8y = 5

13y= 5 + 21

13y = 26

y = 26 / 13

y = 2

Here we got exact value of y. Put this value of y in any of the given equation to get the exact values of x.

**Taking first equation**;

2x – 6y = -6

2x – 6(2) = -6

2x – 12 = -6

2x = -6 + 12

2x = 6**x = 3**

Hence, **(3, 2) is the solution of given linear equation. **