 # Solution of a Linear Equation in Two Variables

Method of substitution: We know that in linear equations involving two variables we need at least two equations in the same unknown variables to find out the values of the variables. In the method of substitution, we find out the value of any one variable from any one of the given equations and substitute that value in the second equation to solve for the value of the variable. This can be better understood with the help of an example.

1. Solve for ‘x’ and ‘y’

2x + y = 9 .................. (i)

x + 2y = 21 .................. (ii)

Solution:

Using method of substitution:

From equation (i) we get,

y = 9 - 2x

Substituting value of ‘y’ from equation (i) in equation (ii):

x + 2(9 – 2x) = 21

⟹ x + 18 – 4x = 21

⟹ -3x = 21 – 18

⟹ -3x = 3

⟹ -x = 1

⟹ x = -1

Substituting x = -1 in equation 2:

y = 9 – 2(-1)

= 9 + 2

= 11.

Hence x = -1 and y = 11.

This method is known as a method of substitution.

Method of elimination: Method of elimination is the method of finding out variables from the equations involving two unknown quantities by eliminating one of the variables and then solving the resulting equation to get the value of one variable and then substituting this value into any one of the equations to get the value of another variable. The elimination is done by multiplying both the equations with such a number that any of the coefficients may have a multiple in common. To understand the concept in a better way, let’s have a look at the example:

1. Solve for ‘x’ and ‘y’:

x + 2y = 10 .......... (i)

2x + y   = 20 ........... (ii)

Solution:

Multiplying equation (i) by 2, we get;

2x + 4y = 20 ......... (iii)

Subtracting (ii) from (iii), we get

4y – y = 0

⟹ 3y = 0

⟹ y = 0

Substituting y = 0 in (i), we get

x + 0 = 10

x = 10.

So, x = 10 and y = 0.

Hope, your query has been solved. If you have more to ask, feel free to write in the comment box.