A system of equations is a group of two or more equations that share variables. In order to solve a system of equations, you need to have at least one equation for every variable. So if there are three variables in the system (x, y and z), then we need to have at least three equations in order to solve.

There are several different methods used to solve a system of equations, including substitution, elimination, and addition. Let’s start by looking at an example of a system of equations:

x - 5y = 12
x + 3y = 4

Addition Method

The first step is to make sure that the variables are neatly lined up under each other (as shown above). Next, we want to decide which variable (x or y) will be easier to eliminate. In order to eliminate a variable, the numbers in front of the variables (the coefficients) must add up to 0. In this particular example, it appears that x is the easier variable to eliminate:

x - 5y = 12
x + 3y = 4

In order to eliminate the x variable, the coefficients must add up to 0. Let’s multiply the second equation by -1 to make this happen:

-1(x + 3y = 4) becomes -x – 3y = -4

Now add the two equations together to eliminate x:

 x - 5y = 12
-x - 3y = -4
------------
- 8y = 8

Solve this simple equation for y:

-8y = 8
y = 8/(-8)
y = -1

Finally, let’s go back and insert the value of y into the original equation to solve for x:

x – 5y = 12
x – 5(-1) = 12
x + 5 = 12
x = 12 – 5
x = 7

So the answer to this system of equations is x=7, y=-1.

Check: To check your work, substitute x=7 and y=-1 into the original equations:

x - 5y = 12
7 – 5(-1) = 12
7 + 5 = 12
12 = 12 Correct!

x + 3y = 4
7 + 3(-1) = 4
7 – 3 = 4
4 = 4 Correct!

Since both equations produce true results when the values are inserted, our solution of x=7, y=-1 is correct.