# SOLVING LINEAR EQUATIONS IN TWO VARIABLES USING ELIMINATION METHOD

Solving Linear Equations in Two Variables Using Elimination Method :

In this section, we will see some example problems using the concept elimination method.

General form of linear equation in two variables is ax + by + c =  0

Procedure for elimination method :

• Using the elimination method, we can  eliminate any one of the variables by combining both equations.
• We can do elimination if and only if the coefficients are same. If they are different, then we have to make the coefficients as same and then we may continue elimination.
• If the coefficients along with the signs are same, we may do elimination by subtracting both the equations.
• If the signs are different and coefficients are same, then we may do elimination by adding both the equations.
• By applying the value of variable we get from the previous step in either of the equations, we will get the value of the remaining variable.

The flow chart given below will help us to understand the procedure better. ## Simultaneous Equations Elimination Method - Examples

Question 1 :

Solve the following system of linear equations by elimination method

13x + 11y  =  70 ,  11x + 13y  = 74

Solution :

13x + 11y  =  70   -------- (1)

11x + 13y  =  74 -------- (2)

By considering the given equations,

Coefficient of x in (1)  =  Coefficient of y in (2)

Coefficient of y in (1)  =  Coefficient of x in (2)

(1) + (2)

24 x + 24 y = 144

Divide the entire equation by 24, we get

x + y  =  6  ------- (3)

(1) - (2)

2x – 2y  =  -4

By dividing the entire equation by 2, we get

x – y  =  -2------- (4)

x +  y  =  6

x –  y  = -2

---------------

2 x = 4

x = 2

Substitute x = 2 in the (3)

2 + y  =  6

y  =  6 – 2

y  =  4

Hence the solution is (2, 4).

Verification :

13 x + 11 y  =  70

13(2) + 11(4)  =  70

26 + 44  =  70

70  =  70

Question 2 :

Solve the following system of linear equations by elimination method

65x – 33y  =  97 and 33x – 65y  =  1

Solution :

65x – 33y  =  97  ------- (1)

33x – 65y  =  1  ---------(2)

By considering the given equations,

Coefficient of x in (1)  =  Coefficient of y in (2)

Coefficient of y in (1)  =  Coefficient of x in (2)

(1) + (2)

98x - 98y  =  98

By dividing the entire equation by 98

x - y  =  1  ------- (3)

(1) - (2)

32x + 32y  =  96

By dividing the entire equation by 32, we get

x + y = 3  ------- (4)

x -  y  =  1

x + y  =  3

---------------

2 x  =  4

x  =  2

Substitute x = 2 in (3)

2 - y  =  1

- y  =  1 – 2

y  =  1

Hence the solution is (2, 1).

Verification :

65 x – 33 y  =  97

65 (2) - 33 (1)  =  97

130 - 33  =  97

97  =  97 After having gone through the stuff and examples,  we hope that the students would have understood, how to solve pair of linear equations by using elimination method.

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