# Solution Of Linear Equation In Two Variable

The solution of a linear equation in two variables x and y, means a pair of values, one for x and one for y which satisfy the given equation.

For example, in equation 3x + 2y = 12, putting the value of x = 2 and y = 3, we get

3x + 2y = (3 × 2) + (2 × 3) = 12

Hence x = 2 and y = 3 is a solution of the above equation. This solution is written as an ordered pair (2, 3).

A linear equation in two variables has infinitely many solutions. To get many solutions of a linear equation in two variables, a variety of values of x is chosen and put in the linear equation in two variables to reduce it in a linear equation in one variable, by solving which the corresponding value of y is obtained. Choosing many different values of x and putting them in the equation many corresponding values of y can be obtained.

Example 1:

Write four solutions for the equation 2x + y = 9.

Solution:

2x + y = 9    =>    y = 9 – 2x

Putting x = 0, we get y = 9 – 2(0) = 9 – 0 = 9

Putting x = 1, we get y = 9 – 2(1) = 9 – 2 = 7

Putting x = 2, we get y = 9 – 2(2) = 9 – 4 = 5

Putting x = 3, we get y = 9 – 2(3) = 9 – 6 = 3

Hence four solutions are (0, 9), (1, 7), (2, 5) and (3, 3).

Example 2:

Check whether (4, 0) and (1, 1) are solutions of equation x – 2y = 4 or not.

Solution:

(4, 0)

Putting x = 4 and y = 0 in equation x – 2y = 4, we get

x – 2y = 4 – 2(0) = 4

Hence (4, 0) is a solution of x – 2y = 4.

(1, 1)

Putting x = 1 and y = 1 in equation x – 2y = 4, we get

x – 2y = 1 – 2(1) = -1 which is not 4.

Hence (1, 1) is not a solution of x – 2y = 4.

Example 3:

Find the value of a so that the equation 5x + 2ay = 3a may have x =1, y = 1 as a solution.

Solution:

If x = 1, y = 1 is a solution, then

 5(1) + 2a(1) = 3a => 5 + 2a = 3a => 5 = 3a – 2a => a = 5     (Answer)

Example 4:

Show that x = 1, y = 1 as well as x = 2, y = 5 is a solution of 4x – y – 3 = 0.

Solution:

4x – y – 3 = 0    =>    4x – y = 3              (1)

Putting x = 1, y = 1 in equation (1), we get

4x – y = 4(1) – 1

= 4 – 1 = 3 (which satisfies equation(1) )

Hence x = 1, y = 1 is a solution of equation (1)

Now putting x = 2, y = 5 in equation (1), we get

4x – y = 4(2) – 5

= 8 – 5 = 3

Hence x = 2, y = 5 is a solution of equation (1).

Hence it is proved that x = 1, y = 1 as well as x = 2, y = 5 is a solution of 4x – y – 3 = 0.

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