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:

y=3-4x
-3x-y=2

Elimination Method

To use the elimination method, you must have a group of two or more equations that share variables (as mentioned above). The idea behind the elimination method is to add or subtract the equations in order to eliminate one of the variables.

If you look closely at the above system, you’ll see that the first equation contains a y and the second equation contains a –y. This is a good candidate for the elimination method because the y variables will cancel each other out. Let’s take a look:

First, move all of the variables to the left-hand side of the equations using alphabetical order:

4x+y=3
-3x-y=2

Now that the equations have been re-ordered, we can “add” the equations together to remove one of the variables (y). At times it may be necessary to multiply each term in the equation by a constant in order to make the coefficients match, but in this case we can just add the equations together without additional calculations:

4x + 1y = 3
-3x - 1y = 2
------------
1x + 0y = 5

x=5

We now have the first half of the problem solved, we must next solve for the other variable (y). In order to solve for y, take the value we calculated for x and substitute it back into either one of the original equations (we selected the first equation for this example):

y=3-4x
y=3-4(5)
y=3-20
y=-17

And that’s it! The solution to this system of equations is x=5, and y=-17. This represents the intersection of the two lines at point (5,-17) when the equations are graphed.