Strategies for Solving Systems of Equations on the ACT

A system of equations is a set of two or more equations that include two or more variables. To solve a system of equations on the ACT Math test, you need one equation for every variable in the system. This usually means two equations and two variables.

You can solve a system of linear equations in two ways:

  • With substitution. With this technique, you solve one equation for a variable in terms of the other(s), and then you substitute this value into the second equation.

  • By combining equations (elimination). To use this method, you add or subtract the two equations in such a way that one variable drops out of the resulting equation.

Both of these methods are similar in that they allow you to write a single equation in one variable, which you can then solve using your usual bag of algebra tricks. After you know the value of one variable, you can substitute this value back into one of the original two equations (usually the easier one) to get the value of the remaining variable.

Substitution is easier to use when a variable in one equation is already isolated or when it can be isolated easily.

Example 1

If x + 9 = y and 7x – 2 = 2y, what is the value of xy?

(A)    48

(B)    49

(C)    50

(D)    51

(E)    52

This question gives you two equations in two variables. In the first equation, y is already isolated on one side of the equation, so substitution should work well. Substitute x + 9 for y in the second equation:

image0.png

Simplify and solve:

image1.png

Now that you know the value of x, substitute this value back into the equation that looks easiest to work with — in this case, the first equation — and solve for y:

image2.png

Thus, x = 4 and y = 13, so xy = 52. The correct answer is Choice (E).

The technique of combining equations is easier to use when both equations contain essentially the same term. Check out the following example.

Example 2

If 4s + 5t = 9 and 9s + 5t = –11, what is the value of s + t?

(A)    2

(B)    1

(C)    0

(D)    –1

(E)    –2

Answering this question using substitution would be difficult because neither variable is very easy to isolate on one side of the equations. However, both equations include the term 5t, so you can combine the two equations using subtraction.

image3.png

When you subtract one equation from the other, the t term drops out. The resulting equation is easy to solve:

image4.png

As always, when you know the value of one variable, you can substitute this value back into either equation — whichever looks easiest — and solve for the other variable, like this:

image5.png

So s = –4 and t = 5, meaning s + t = 1. As a result, the correct answer is Choice (B).

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