Math Core

Last year you solved equations like $x + 9 = 15$ and $4x = 28$ in one step. Many real equations take two steps, such as $4x + 9 = 37$ . The good news: you already have every tool you need. You just have to use them in the right order.

What makes an equation "two-step"

In $4x + 9 = 37$ , two things happened to $x$ :

It was multiplied by $4$ .
Then $9$ was added.

To get $x$ alone, you undo those steps in reverse order. Think about getting dressed: you put on socks, then shoes. To undress, you take off the shoes first, then the socks. The last thing done to $x$ is the first thing you undo.

Solving a two-step equation

Undo the addition or subtraction first (add or subtract on both sides).
Then undo the multiplication or division (multiply or divide both sides).
Check by substituting your answer into the original equation.

Whatever you do to one side, do the same to the other side.

Worked example: Multiply, then add

Solve $4x + 9 = 37$ .

The last step done to $x$ was adding $9$ , so subtract $9$ first. Then divide by $4$ .

\begin{aligned} 4x + 9 &= 37 \\ 4x + 9 - 9 &= 37 - 9 \\ 4x &= 28 \\ \dfrac{4x}{4} &= \dfrac{28}{4} \\ x &= 7 \end{aligned}

Check: $4(7) + 9 = 28 + 9 = 37$ . ✓

Worked example: Divide, then subtract

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

First add $3$ to both sides. Then multiply both sides by $5$ .

\begin{aligned} \dfrac{x}{5} - 3 &= 4 \\ \dfrac{x}{5} &= 7 \\ x &= 7 \times 5 = 35 \end{aligned}

Check: $35 \div 5 - 3 = 7 - 3 = 4$ . ✓

Negative numbers

Equations with negative numbers follow the same two steps. Keep the sign attached to the number that follows it. In $12 - 3x$ , the term with $x$ is $-3x$ , so $x$ is multiplied by $-3$ .

Worked example: A negative coefficient

Solve $12 - 3x = 27$ .

Subtract $12$ from both sides. Then divide both sides by $-3$ .

\begin{aligned} 12 - 3x &= 27 \\ -3x &= 15 \\ \dfrac{-3x}{-3} &= \dfrac{15}{-3} \\ x &= -5 \end{aligned}

Check: $12 - 3(-5) = 12 + 15 = 27$ . ✓

Common mistake

Watch the sign in front of the variable. In $12 - 3x = 27$ , many students divide by $3$ instead of $-3$ and get $x = 5$ . Check it: $12 - 3(5) = -3$ , not $27$ . The minus sign belongs to the $3$ .

Equations with parentheses

An equation like $4(x + 6) = 36$ has the form $p(x + q) = r$ . Here $6$ was added first, and then the sum was multiplied by $4$ . You can solve it two ways.

Worked example: Two ways to solve

Solve $4(x + 6) = 36$ .

Way 1: undo the multiplication first. Divide both sides by $4$ , then subtract $6$ .

\begin{aligned} 4(x + 6) &= 36 \\ x + 6 &= 9 \\ x &= 3 \end{aligned}

Way 2: distribute first. Use the distributive property, then solve the two-step equation.

\begin{aligned} 4x + 24 &= 36 \\ 4x &= 12 \\ x &= 3 \end{aligned}

Both ways give $x = 3$ . Check: $4(3 + 6) = 4(9) = 36$ . ✓

Dividing first is often quicker when the right side divides evenly. Distributing first works every time.

Fractions and decimals

The steps stay the same with fractions and decimals. To undo multiplying by a fraction like $\dfrac{2}{3}$ , multiply both sides by its reciprocal, $\dfrac{3}{2}$ .

For example, to solve $\dfrac{2}{3}m + 1 = 9$ , subtract $1$ to get $\dfrac{2}{3}m = 8$ . Then $m = 8 \times \dfrac{3}{2} = 12$ .

Tip

Checking takes ten seconds and catches almost every mistake. Always substitute into the original equation, not one of your later steps, because a mistake in an early step would carry through.