One-Step Equations: How to Solve Them

A one-step equation takes just one operation to solve. To solve one-step equations, undo the operation on the variable with its inverse: subtract to undo adding, add to undo subtracting, divide to undo multiplying, and multiply to undo dividing. Do the same thing to both sides. For example, $x + 7 = 12$ gives $x = 5$ .

Before this lesson: You should know the difference between an expression and an equation and how to add and subtract negative numbers.

Extra practice: The Algebra 1 Unit 2 Practice Pack has printable problems, answer keys and a cheat sheet for this unit.

What Is a One-Step Equation?

An equation says two things are equal. Solving it means finding the value of the variable that makes it true. In a one-step equation, only one operation is done to the variable, like $x + 3 = 7$ or $5x = 30$ .

Think of an equation as a balance scale. Both sides weigh the same. If you take something off one side, you must take the same amount off the other side, or the scale tips.

Two balance scales. The first shows x + 3 = 7: a box labeled x and 3 circles on the left, 7 circles on the right. After taking 3 circles off each side, the second scale shows x = 4. — Take 3 off both sides and the scale stays balanced: x + 3 = 7 becomes x = 4.

Your goal is to get the variable by itself on one side. This is called isolating the variable. Here, taking 3 off both sides leaves $x = 4$ .

How to Solve One-Step Equations

Each operation has an inverse operation that undoes it. Use the inverse on both sides of the equation.

If the equation has	Do this to both sides	Example
$x + a = b$	Subtract $a$	$x + 7 = 12$ , so $x = 12 - 7 = 5$
$x - a = b$	Add $a$	$x - 4 = 9$ , so $x = 9 + 4 = 13$
$ax = b$	Divide by $a$	$5x = 30$ , so $x = \frac{30}{5} = 6$
$\frac{x}{a} = b$	Multiply by $a$	$\frac{x}{3} = 8$ , so $x = 8 \cdot 3 = 24$

Then check your answer. Put it back into the original equation. If both sides are equal, your answer is right.

Worked Examples

Example 1: Addition with a negative answer

Solve $x + 9 = 4$ .

x + 9 - 9 = 4 - 9

Step 1. 9 is added to the variable, so subtract 9 from both sides.

x = -5

Step 2. Simplify each side. $9 - 9 = 0$ , and $4 - 9 = -5$ . (See adding and subtracting integers.)

Answer: $x = -5$

Check: $-5 + 9 = 4$ . It works.

Example 2: Subtraction

Solve $y - 6 = -2$ .

y - 6 + 6 = -2 + 6

Step 1. 6 is subtracted from the variable, so add 6 to both sides.

y = 4

Step 2. Simplify each side. $-6 + 6 = 0$ , and $-2 + 6 = 4$ .

Answer: $y = 4$

Check: $4 - 6 = -2$ . It works.

Example 3: Multiplication by a negative

Solve $-4n = 28$ .

\frac{-4n}{-4} = \frac{28}{-4}

Step 1. The variable is multiplied by $-4$ , so divide both sides by $-4$ .

n = -7

Step 2. Simplify each side. A positive divided by a negative is negative. (See multiplying and dividing integers.)

Answer: $n = -7$

Check: $-4(-7) = 28$ . It works.

Example 4: A fraction coefficient

Solve $\frac{2}{3}x = 10$ .

\frac{3}{2} \cdot \frac{2}{3}x = \frac{3}{2} \cdot 10

Step 1. The variable is multiplied by $\frac{2}{3}$ . To undo it, multiply both sides by the reciprocal, $\frac{3}{2}$ . The reciprocal is the fraction flipped over. (See multiplying and dividing fractions.)

x = \frac{3}{2} \cdot 10

Step 2. Simplify the left side. $\frac{3}{2} \cdot \frac{2}{3} = 1$ .

x = 15

Step 3. Multiply on the right. $\frac{3}{2} \cdot 10 = \frac{30}{2} = 15$ .

Answer: $x = 15$

Check: $\frac{2}{3}(15) = 10$ . It works.

Example 5: Word problem

Concert tickets cost $12 each. A group spends $84 on tickets. How many tickets did they buy?

12t = 84

Step 1. Let $t$ = the number of tickets. The cost is $12t$ , and it equals 84.

\frac{12t}{12} = \frac{84}{12}

Step 2. Divide both sides by 12.

t = 7

Step 3. Simplify each side.

Answer: They bought 7 tickets.

Check: 7 tickets at $12 each cost $12(7) = 84$ dollars. It works.

Common Mistakes

Mistake	What it looks like	How to fix it
Changing only one side	$x + 5 = 12$ becomes $x = 12$	Subtract 5 from both sides: $x = 7$ .
Using the same operation, not the inverse	$x - 4 = 10$ , so $x = 6$	Undo subtracting by adding: $x = 14$ .
Losing the negative when dividing	$-3x = 15$ , so $x = 5$	Divide by $-3$ : $x = -5$ .
Multiplying instead of dividing	$4x = 20$ , so $x = 80$	Divide by 4: $x = 5$ .
Forgetting that $-x$ means $-1x$	$-x = 8$ , so $x = 8$	Divide both sides by $-1$ : $x = -8$ .

Quick check: Always put your answer back into the original equation. It takes ten seconds and catches most mistakes.

One-Step Equations with Decimals and Fractions

The steps don’t change when the numbers are decimals or fractions. Only the arithmetic does. If you need a refresher, see operations with decimals.

\begin{aligned} x + 2.5 &= 7.1 & &\Rightarrow & x &= 7.1 - 2.5 = 4.6 \\ x - \tfrac{3}{4} &= \tfrac{1}{2} & &\Rightarrow & x &= \tfrac{1}{2} + \tfrac{3}{4} = \tfrac{5}{4} \end{aligned}

Practice Problems

Solve each equation. Check your answer.

$x + 12 = 20$
$a - 7 = -3$
$15 = y + 22$
$9k = -63$
$-6p = -42$
$\frac{n}{4} = 11$
$\frac{w}{-3} = 5$
$\frac{3}{4}x = 12$
$-x = 13$
A rope is cut into 6 equal pieces. Each piece is 2.5 meters long. Write and solve an equation to find the length of the rope.

Answers

Show answers

x + 12 - 12 = 20 - 12

Subtract 12 from both sides.

x = 8

Simplify.

Check: $8 + 12 = 20$

a - 7 + 7 = -3 + 7

Add 7 to both sides.

a = 4

Simplify.

Check: $4 - 7 = -3$

15 - 22 = y + 22 - 22

Subtract 22 from both sides.

-7 = y

Simplify.

y = -7

Write the variable first.

Check: $-7 + 22 = 15$

\frac{9k}{9} = \frac{-63}{9}

Divide both sides by 9.

k = -7

Simplify.

Check: $9(-7) = -63$

\frac{-6p}{-6} = \frac{-42}{-6}

Divide both sides by −6.

p = 7

Simplify. A negative divided by a negative is positive.

Check: $-6(7) = -42$

4 \cdot \frac{n}{4} = 4 \cdot 11

Multiply both sides by 4.

n = 44

Simplify.

Check: $\frac{44}{4} = 11$

-3 \cdot \frac{w}{-3} = -3 \cdot 5

Multiply both sides by −3.

w = -15

Simplify.

Check: $\frac{-15}{-3} = 5$

\frac{4}{3} \cdot \frac{3}{4}x = \frac{4}{3} \cdot 12

Multiply both sides by the reciprocal, 4/3.

x = \frac{4}{3} \cdot 12

Simplify the left side.

x = 16

Multiply.

Check: $\frac{3}{4}(16) = 12$

\frac{-x}{-1} = \frac{13}{-1}

$-x$ means $-1x$ . Divide both sides by −1.

x = -13

Simplify.

Check: $-(-13) = 13$

10.

\frac{L}{6} = 2.5

Let $L$ = the length of the rope. Each piece is the rope divided by 6.

6 \cdot \frac{L}{6} = 6 \cdot 2.5

Multiply both sides by 6.

L = 15

Simplify.

Check: $\frac{15}{6} = 2.5$
The rope is 15 meters long.

Want more practice? The Algebra 1 Unit 2 Practice Pack has 51 pages of problems on this and the other lessons in the unit, with step-by-step answers and a cheat sheet.

One-Step Equations FAQ

What is a one-step equation?

It is an equation you can solve with one inverse operation, such as $x - 8 = 3$ or $6x = 42$ .

How do you solve one-step equations?

Find the operation done to the variable. Then do the inverse operation to both sides, so the variable is alone on one side.

What is an inverse operation?

It is the operation that undoes another one. Addition and subtraction undo each other, and multiplication and division undo each other.

How do you check your answer?

Replace the variable with your answer in the original equation. If both sides come out equal, the answer is correct.

Summary

A one-step equation needs one inverse operation to solve.
Whatever you do to one side, do to the other side.
Add to undo subtracting, subtract to undo adding, multiply to undo dividing, and divide to undo multiplying.
To undo a fraction coefficient, multiply by its reciprocal.
Check your answer by putting it back into the original equation.

Previous lesson: Distributive Property

Next lesson: Two-Step Equations

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