Combination Method for Solving Math Problems

Instructor: Michael Quist

Michael has taught college-level mathematics and sociology; high school math, history, science, and speech/drama; and has a doctorate in education.

There are three common methods for for solving systems of linear equations. In this lesson, we'll discuss the combination method, which involves adding equations together to eliminate variables.

What is the Combination Method for Solving Math Problems?

Just looking at a stack of multi-variable equations and imagining trying to solve them can drive you crazy, but once you know the procedures you can always get to an answer, or discover that there aren't any.

The combination method of solving systems of equations is a way of adding equations together in such a way that the variables are set aside, one by one. Finally, when only one variable remains in the equation, you can learn its value. Then you can plug that value in, which simplifies all the rest of the equations. Let's figure out how to do it!

Two Equations with Two Variables

First, let's solve a system of equations that has two equations and two variables. For example, say Susan and Joanne go to the store to buy music and movie DVDs. You want to find out how many of each item they bought, and have figured out the following:

  • a music DVD costs $14
  • a movie DVD costs $21
  • they spent a total of $217
  • they bought 12 items

Since there are two variables (the number of music DVDs and the number of movie DVDs), we'll need two equations. The first one is easy. The girls bought 12 items altogether, so that means that the number of music DVDs plus the number of movie DVDs will equal 12. We can write that in an equation by assigning x to the number of music DVDs and y to the number of movie DVDs.

x + y = 12

The second equation takes a little more thought. We know how much each kind of item costs, and we know the total that the girls spent, so we should be able to use that plus our x and y variables to show the cost relationship. Since the music DVDs cost $14 each, we can express the total spent on music DVDs as 14x. Similarly, we can express the movie DVD cost as 21y. Adding them together, and assigning the total cost, we can express the relationship with an equation.

14x + 21y = 217

Now it's time to use our combination method. What we're going to do is add the two equations together to get rid of one of the variables. Obviously, that won't happen if we add them together as they are:

x + y = 12

14x + 21y = 217

The x terms will add up to 15x, the y terms add up to 22y, and the constants will add up to 229.

15x + 22y = 229

Well, that wasn't much help! We just have a new equation, but we still have two variables staring at us. Time for a trick. If we want to eliminate the y variable, for example, we'll need the two y terms to cancel. We can arrange that by multiplying both sides of the first equation by -21.

-21x - 21y = -252

14x + 21y = 217

What will happen if we add them together now?

-7x = -35

x = 5 (divide both sides by -7)

If we plug that value in for x in our first equation, we'll get a solution.

5 + y = 12 (plug in our value of x)

y = 7 (subtract 5 from both sides of the equation).

The girls bought 5 music DVDs and 7 movie DVDs.

Three Equations with Three Variables

That wasn't too bad, but it gets a lot crazier when you go to three equations and three variables. Let's redefine the shopping trip. Now they're buying posters, as well as their music and movie DVDs. Here are the facts:

Say our girls in the first problem are buying music DVDs for $12, movie DVDs for $20, and posters for $5. They buy 23 items and spend $240. You also know that

  • a music DVD costs $12
  • a movie DVD costs $20
  • a poster costs $5
  • they spent a total of $240
  • they bought 23 items
  • they bought twice as many posters as movie DVDs

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