What is the solution to the system of equations shown below?

6x – 3y = -12

4x + 3y = 2

A) (2,2)

B) (-2,-1)

C) (-1,-1)

D) (-1,2)

E) (1,-2)

In order to solve this problem, we first need to understand what it’s asking. So, what is a “system of equations”? Well, that simply means that we have more than one equation that deals with the same variables. Don’t let that name overwhelm you into thinking it’s more complicated than it really is! In this case, we have to linear equations and solving a system of linear equations simply means finding the point where the two lines intersect. At that point you’ll get an x-value and a y-value that work in both equations.

There are several different ways to solve a system of linear equations and trick is to figure out which one is most efficient for the equations that you’re given. The three methods are:

1. Graphing- If you turn your equations into y=mx+b form you should be able to graph them out and figure out where they intercept. The advantage of this method is that it’s very straight-forward. The disadvantage is that it requires you to be very precise with the lines you draw and can be infeasible when you have an intersection that’s relatively far from the origin. Graphing is typically best used to estimate, or verify a solution.

2. Substitution- This method helps you turn unruly two-variable equations into single-variable equations that can easily be solved. You do this by finding one variable in terms of the other. Let’s see how that works:

Take the first equation.

6x – 3y = -12

Solve for one of the variables.

3y = 6x + 12

y= 2x +4

Then, plug that value into the second equation.

4x + 3(2x+4) = 2

4x+ 6x+ 12 = 2

10x = -10

x = -1

Now that you have the value for x, you can plug that back into either one of your equations to find the value for y.

4(-1) +3y = 2

3y= 6

y= 2

The great part about substitution is that you know it’s going to work. The downside is that there are quite a few steps in the process so it can be somewhat time consuming. That’s where our third method comes in.

3. Combination- This method involves adding or subtracting the two equations with each other with the goal of eliminating one of the variables. When combination can be done without too much manipulation, it’s the easiest and quickest way to solve a system of equations. That’s the case here.

6x – 3y = -12

+ 4x + 3y = 2

________________

10x = -10

As you can see here, when we add the two equations together the y variable drops out and you have the value for x so you can solve from there.

Remember not to worry when you see a system of equations. Choose the best method to solve based on the problem you’re given and you’re all set!