One-Step Equations
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.
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.
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.
Step 1. 9 is added to the variable, so subtract 9 from both sides.
\begin{aligned}
x + 9 &= 4 \\
x + 9 - 9 &= 4 - 9 \\
x &= -5
\end{aligned}Check: -5 + 9 = 4. It works.
Answer: x = -5
Example 2: Subtraction
Solve y - 6 = -2.
Step 1. 6 is subtracted from the variable, so add 6 to both sides.
\begin{aligned}
y - 6 &= -2 \\
y - 6 + 6 &= -2 + 6 \\
y &= 4
\end{aligned}Check: 4 - 6 = -2. It works.
Answer: y = 4
Example 3: Multiplication by a negative
Solve -4n = 28.
Step 1. The variable is multiplied by -4, so divide both sides by -4.
\begin{aligned}
-4n &= 28 \\
\frac{-4n}{-4} &= \frac{28}{-4} \\
n &= -7
\end{aligned}Check: -4(-7) = 28. It works.
Answer: n = -7
Example 4: A fraction coefficient
Solve \frac{2}{3}x = 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.
\begin{aligned}
\tfrac{2}{3}x &= 10 \\
\tfrac{3}{2} \cdot \tfrac{2}{3}x &= \tfrac{3}{2} \cdot 10 \\
x &= 15
\end{aligned}Check: \frac{2}{3}(15) = 10. It works.
Answer: x = 15. For more on reciprocals, see multiplying and dividing fractions.
Example 5: Word problem
Concert tickets cost $12 each. A group spends $84 on tickets. How many tickets did they buy?
Step 1. Let t = the number of tickets. The cost is 12t, so the equation is 12t = 84.
Step 2. Divide both sides by 12.
t = \frac{84}{12} = 7Check: 7 tickets at $12 each cost 12(7) = 84 dollars. It works.
Answer: They bought 7 tickets.
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
- Subtract 12: x = 8
- Add 7: a = 4
- Subtract 22: 15 - 22 = y, so y = -7
- Divide by 9: k = -7
- Divide by -6: p = 7
- Multiply by 4: n = 44
- Multiply by -3: w = -15
- Multiply by \frac{4}{3}: x = \frac{4}{3}(12) = 16
- Divide by -1: x = -13
- Let L = the length of the rope. \frac{L}{6} = 2.5, so L = 2.5(6) = 15. The rope is 15 meters long.
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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