Solution of a Linear Equation in Two Variables

Earlier we have studied about the linear equations in one variable. We know that in linear equations in one variable, only one variable is present whose value we need to find out by doing calculations that involve simple operations such as +,-,/ and *. Also, we are aware that only one equation is sufficient for finding out the value of the variable as there is only one variable is present.

The concept of the linear equations remains unchanged in the case of linear equations in two variables also. The thing that changes is that there are two variables present in this case instead of one variable and other thing that changes is the methods of solving the equations to find out the values of the unknown quantities. Also, atleast two equations are required to solve the linear equations involving two unknown quantities. 

ax + by = c and ex + fy = g

are the two equations with linear equations in two variables with a, b, c, d, e and f as constants and ‘x’ and ‘y’ as the variables whose values we need to calculate.

Mostly, there are two methods which are used to solve such equations involving two variables. These methods are:

I. Method of substitution, and

II. Method of elimination.

Method of substitution: We know that in linear equations involving two variables we need atleast two equations in same unknown variables to find out the values of the variables. In the method of substitution we find out 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)


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 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 value of one variable and then substituting this value into anyone 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)


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.

9th Grade Math

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