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 – 4d

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

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