Let’s go over the process to solve a system of equations. There are two popular ways of solving a system of equations: Substitution & Elimination. For those of you with graphing calculators, if you are lucky enough to get a question like this in the non-calculator section, you can plug both equations into your calculator and solve the system graphically. Between Substitution & Elimination, students usually have a preference; however, it’s best to let the question make that decision for you. For example, if you are presented with a problem like the one below, it is probably best to use Elimination. If a problem has an equation where the *x* or the *y* is isolated on one side of the equal sign, it is probably best to use Substitution.

3*x* + 4*y* = -23

2*y* – *x* = -19

What is the solution to the system of equations above?

Let’s solve this question using Elimination.

First, let’s rearrange the terms in the second equation to match the order in the first equation:

3*x* + 4*y* = -23

–*x* + 2*y* = -19

Typically, we want to eliminate the variable that the question does not ask for (if the question is only asking for either *x* or *y*). Since this question asks for the solution to the system, it does not matter which variable we eliminate first. Let’s multiply the second equation by to eliminate the *y*:

-2(-*x* + 2*y* = -19)

2*x* – 4*y* = 38

Now let’s add the two equations together & solve for *x*:

3*x* + 4*y* = -23

+(2*x* – 4*y* = 38)

5*x* = 15

*x* = 3

Now that we have figured out the value of *x*, let’s plug it back into one of the equations to get the value of *y*:

3(3) + 4*y* = -23

9 + 4*y* = -23

4*y* = -32

*y* = -8

The solution for this system of equations is (3, -8).