# Elimination Method

The elimination method is one of the methods to solve pair of simultaneous linear equations. By applying arithmetic properties, we reduce one of the equations that has only one variable and determine the another one. This section will help you to understand how to use elimination method to solve linear equations with two variables in a simple way.
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## What is Elimination Method

A method can be used to solve the pair of linear equations with two variables having degree one.  This method is known as the “Gaussian elimination method."

## Elimination Method for Solving Linear Systems

In the elimination method, the very first step is to obtain an equation in one variable either by adding or subtracting the equations. If again variables are not eliminated, we multiply one or both of the equations with a coefficients to get an equivalent linear system. Now it is easy to eliminate one of the variables.

### How to solve Linear Equations by Elimination

While solving Linear equations of two variables by the method of elimination, we follow the following steps.
Step 1:Number the two equations as (1) and (2).
Step 2: Multiply the two equations by the suitable numbers so that the coefficients of one of the variables is same.
Step 3: Add or subtract the two equations so that one of the variable is eliminated and we obtain the linear equation of one variable.
Step 4: Solve for the variable obtained from the equation in step 3.
Step 5: Substitute the value of this variable in any one of the two equations, so that we get a linear equations of the other variable.
Step 6: Solve for the other variable obtained in Step 5.
Step 7: Express the values of x and y obtained from Step 3 and step 6 in the form ( x, y) = (  ___ , ___  ).

## Examples

Let us solve some of the examples where pair of linear equations are solved using elimination method:
Example 1: Solve 2y - 3x = 4 and y- 4x = 3

Solution:
Step 1
: 2y - 3x = 4   ....(1) and
y - 4x = 3                .....(2)

Step 2: Multiply equation (2) by 2 so that coefficients of y are same for both the equations.
-2(y - 4x = 3)
New equations are:
-3x + 2y = 4 and
8x - 2y = -6

Step 3:
Add these equations to eliminate y: This implies, 5x = -2

Step 4: Solve for x:
x = -2/5
Plug it back the value of x in any of the given equations to solve for y.

Step 5: Choose: 2y - 3x = 4
By substituting the value of x, we get
2y - 3(-2/5) = 4
2y + 6/5 = 4
y = 7/5

The solution set of the given linear equations is (x, y) = (-2/5, 7/5). Answer!

Example 2: Solve x + 3y = 10 and x + 2y = -5 using elimination method.
Solution: Let us solve the given system of equations by elimination method
x + 3y = 10; x + 2y = -5

x + 3y = 10  equation (1)
x + 2y = -5  equation (2)

Since the coefficient of x is same for the both the equations. Subtract equation 2 form equation 1, So y = 15
Now solve for x:
Substitute 15 for y in x + 3y = 10
x + (3)(15) = 10
x + 45 = 10
x + 45 + (-45) = 10 + (-45)  (Add -45 to both sides)
x = -35

## Practice Problems

Practice below problems to test your skill.
Problem 1: Find x and y: x + y = 1/3 and 2x - 3y = 5
Problem 2: Solve 2x - y = 10 and 3x + 5y = 11
Problem 3: Solve systems of linear equations by elimination method
3x + 7y = 1 and x - 2y = 4