Math Core

A taxi charges a $3 pickup fee plus $2 per mile, and your ride cost $19. How far did you go? That question becomes the equation $2m + 3 = 19$ , where two things happen to $m$ . Solving it takes two inverse operations, done in the right order.

Two operations, reverse order

In $2m + 3 = 19$ , follow what happens to $m$ using the order of operations:

First $m$ is multiplied by $2$ .
Then $3$ is added.

To get $m$ back, undo those steps in reverse order. Imagine wrapping a gift: you put it in a box, then add wrapping paper. To open it, you tear off the paper first, then open the box. The last thing done is the first thing undone.

Solving a two-step equation

Undo the addition or subtraction (the last operation) on both sides.
Undo the multiplication or division on both sides.
Check your answer in the original equation.

Worked example: The taxi ride

Solve $2m + 3 = 19$ .

\begin{aligned} 2m + 3 &= 19 \\ 2m + 3 - 3 &= 19 - 3 \\ 2m &= 16 \\ m &= 8 \end{aligned}

Check: $2(8) + 3 = 16 + 3 = 19$ . ✓ The ride was $8$ miles.

Worked example: Division and a negative answer

Solve $\dfrac{n}{6} + 2 = -1$ .

Subtract $2$ from both sides, then multiply both sides by $6$ .

\begin{aligned} \dfrac{n}{6} + 2 &= -1 \\ \dfrac{n}{6} &= -3 \\ n &= -3 \times 6 = -18 \end{aligned}

Check: $-18 \div 6 + 2 = -3 + 2 = -1$ . ✓

When the variable term is negative

In $9 - 4x = 29$ , the term with $x$ is $-4x$ . The minus sign belongs to the $4$ . It helps to read the equation as $9 + (-4x) = 29$ .

Worked example: A negative coefficient

Solve $9 - 4x = 29$ .

Subtract $9$ from both sides. Then divide by $-4$ .

\begin{aligned} 9 - 4x &= 29 \\ -4x &= 20 \\ x &= \dfrac{20}{-4} = -5 \end{aligned}

Check: $9 - 4(-5) = 9 + 20 = 29$ . ✓

Common mistake

Don't undo the steps in the wrong order. In $2m + 3 = 19$ , dividing by $2$ first means you have to divide every term: $m + 1.5 = 9.5$ . That works, but it's easy to forget the $3$ and write $m + 3 = 9.5$ . Undo the addition or subtraction first and you avoid the trap.

When the operation is on a group

Sometimes the adding happens first, and the result is divided or multiplied. In $\dfrac{x + 5}{3} = 4$ , the fraction bar groups $x + 5$ . So $5$ is added first, then the sum is divided by $3$ . Reverse order means you undo the division first.

Worked example: Undo the grouping

Solve $\dfrac{x + 5}{3} = 4$ .

Multiply both sides by $3$ , then subtract $5$ .

\begin{aligned} \dfrac{x + 5}{3} &= 4 \\ x + 5 &= 12 \\ x &= 7 \end{aligned}

Check: $\dfrac{7 + 5}{3} = \dfrac{12}{3} = 4$ . ✓

Tip

Before you start, say out loud what happens to the variable, in order: "times $2$ , then plus $3$ ." Then read your list backward and do the inverses: "minus $3$ , then divide by $2$ ."