 # Should I use Combination or Substitution?

- Nov 12, 09:00 AM Comments 

You might remember back in the day in Algebra class that you learned how to work with systems of linear equations. These are problems when you have multiple equations and multiple variables, and you have to find a way to solve for each variable. Although we have two variables, we can only solve for one at a time. Therefore, our goal is to eliminate one of the variables, and there are two methods for doing so: Combination and Substitution. Both methods always work, but which should you use?

In short, it depends on the problem. One of the core competencies in Kaplan’s GMAT curriculum is Pattern Recognition, and our courses and materials aim to teach you to recognize which technique will get you to the answer fastest. There are certain “triggers” that let us know whether to combine or substitute; read below for the most fundamental triggers:

Combination is ideal when you can easily eliminate a variable by adding or subtracting the equations. For example, by adding the following equations, you can get rid the $y$’s. $5x - 2y =8$ $3x + 2y= 8$

Thus, $8x=16$, $x=2$, and we are golden. You don’t want to substitute in this situation because it’s hard to get a “clean” value to substitute. Who wants to plug $\frac{8 - 3x}{2}$ in for $y$? Fractions are no fun when you’re pressed for time.

However, we should use substitution if we have a simple coefficient in front of one of the variables. $x + 3y = 20$ $3x-4y = 8$

This problem lends itself to substitution. It’s very easy to isolate x in the first equation (just subtract $3y$ from each side to get $x=20-3y$), so we can substitute $(20-3y)$ for $x$ in the second equation. Doing so, we find that $3(20-3y) - 4y= 8$

Carrying out the algebra, we can find that $y=4$.

Many combination aficionados would still argue that you could multiply the first equation by 3 and subtract the two equations to eliminate the $x$’s.

There are many cases when choosing combination or substitution is merely a matter of personal preference. If you feel more comfortable with one approach, that’s what you should use in these 50/50 situations on Test Day. Still, remember that your favored approach may change as you prep. For example, substitution may initially feels more familiar for most students, but many of them get more comfortable with combination as time goes on.

But, no matter what, you need to be ambidextrous! Practice both approaches so that you can choose the right one on test day.

Ben Leff
Kaplan GMAT 