Solving Linear Equation by Elimination Method

 Elimination method is a way to manipulate systems of equations in order to solve them algebraically. This is actually very similar to the way we manipulate single equations in order to solve them.

The elimination method is a technique for solving systems of linear equations. Let's walk through a couple of examples.


We're asked to solve this system of equations:
3y + 11x = −10
4y − 11x​​​ = 17
We notice that the first equation has a 11x and the second equation has a −11x term. These terms will cancel if we add the equations together—that is, we'll eliminate the x terms:​​
3y + 11x = −10
4y − 11x​​​ = 17
-----------------
7y + 0 = 7

Solving for y, we get:
7y + 0 = 7
7y = 7
y=1
plugging this value back into our first equation, we solve for the other variable:
3y + 11x = −10
3.1 + 11x = -10
11x = -10-3
11x = -13
x = -13/11
The solution to the system is x=−13/11 and y=1

Not all time this system will work easily. Suppose a system like 4x +2y = 24 and −6x +2y = 4. If you add these two equations together, no variables are eliminated. This time we have to multiply any equation with -1.

4x   + 2y    =    24 

 − (− 6x + 2y ) = −(4)
or
4x   +  2y   = 24  

6x  −  2y    = −4
-----------------------
10x + 0y =20

x = 2

Put this value back into our first equation, we solve for the other variable:

4x + 2y =24

4.2 + 2y = 24

8 + 2y = 24 

2y = 24-8

2y = 16

y = 8

The solution to the system is x=2 and y=8