# Solving ACT Word Problems Using a System of Equations

Solving word problems is one of the most common reasons to use a system of equations. For example, some word problems in the ACT Math test that would be difficult to approach using a single variable are relatively easy when you use more than one variable.

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

### Example

Dorian and Micah have been saving money from their summer jobs. If Dorian had twice as much money and Micah had half as much, together they would have $2,075. And if Micah had twice as much money and Dorian had half as much, together they would have $2,300. How much money does Dorian have?

(A) $800

(B) $850

(C) $900

(D) $950

(E) $1,000

You could solve this problem using only one variable, but that approach would be tricky and would likely lead to a mistake along the way. Instead, use two variables, letting *d* equal Dorian’s money and *m* equal Micah’s money. Set up two equations as follows:

To eliminate the fractions, multiply both of these equations by 2:

This system of equations is easy to solve using substitution. Begin by isolating *m* in the first equation:

*m* = 4,150 – 4*d*

Now substitute 4,150 – 4*d* for *m* in the second equation, and then solve for *d*:

Dorian has $800, so the correct answer is Choice (A).