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Solving Systems of Equations with the Elimination Method (Subtraction)

Systems of Equations, where you have more than one equation and are trying to find the intersection between them, can be solved in a number of ways. Here we will look at the Elimination Method, also known as the Addition/Subtraction method.

Let’s look at a simple system of equations:

Equation 1: 3x+5y=12
Equation 2: 7x+5y=8

Notice that both equations have 5y terms. You can eliminate the y-terms from the equations if you add or subtract the equations. If the terms you want to eliminate have the same sign, you need to subtract the equations.

(3x+5y=12)
(7x+5y=8)
___________
-4x + 0 = 4

Notice that 5y – 5y = 0. You have now eliminated y from the system! It’s easy to solve for x now:

-4x = 4
x = 4/-4
x = -1

Now put x=-1 into either equation, and solve for y:

3x +5y = 12
3(-1) +5y = 12
-3 + 5y = 12
5y = 12+3
5y = 15
y = 15/5
y = 3

So the two lines intersect at (-1,3). Here’s a graph of what the system looks like: