One-Step Equations

An equation is a statement that two expressions are equal. Solving an equation means finding the value of the variable that makes the statement true. In this lesson, you will start with the simplest type: one-step equations, where only a single operation separates the variable from the answer.

What Is an Equation?

An equation has three parts: a left side, an equals sign, and a right side.

$x + 5 = 12$

This says “some number plus 5 equals 12.” Your job is to find the number.

The Balance Model

Think of an equation as a balance scale. Whatever you do to one side, you must do to the other side to keep the scale balanced.

• If you add 3 to the left side, add 3 to the right side.
• If you divide the left side by 2, divide the right side by 2.

This rule — do the same thing to both sides — is the foundation of all equation solving.

Inverse Operations

Every operation has an inverse (opposite) that undoes it:

To isolate the variable, use the inverse of whatever operation is being applied to it.

Addition Equations

Example 1: Solve $x + 7 = 15$

The variable $x$ has 7 added to it. Use the inverse — subtract 7 from both sides:

$x + 7 - 7 = 15 - 7$

$x = 8$

Check: $8 + 7 = 15$ — that checks out.

Example 2: Solve $n + 13 = 20$

Subtract 13 from both sides:

$n + 13 - 13 = 20 - 13$

$n = 7$

Check: $7 + 13 = 20$ — confirmed.

Subtraction Equations

Example 3: Solve $y - 9 = 4$

The variable $y$ has 9 subtracted from it. Use the inverse — add 9 to both sides:

$y - 9 + 9 = 4 + 9$

$y = 13$

Check: $13 - 9 = 4$ — correct.

Example 4: Solve $a - 25 = -3$

Add 25 to both sides:

$a - 25 + 25 = -3 + 25$

$a = 22$

Check: $22 - 25 = -3$ — confirmed.

Multiplication Equations

Example 5: Solve $4x = 28$

The variable $x$ is multiplied by 4. Use the inverse — divide both sides by 4:

$\frac{4x}{4} = \frac{28}{4}$

$x = 7$

Check: $4(7) = 28$ — correct.

Example 6: Solve $-3m = 18$

Divide both sides by $-3$:

$\frac{-3m}{-3} = \frac{18}{-3}$

$m = -6$

Check: $-3(-6) = 18$ — confirmed. A negative times a negative is positive.

Division Equations

Example 7: Solve $\frac{x}{5} = 9$

The variable $x$ is divided by 5. Use the inverse — multiply both sides by 5:

$\frac{x}{5} \times 5 = 9 \times 5$

$x = 45$

Check: $\frac{45}{5} = 9$ — correct.

Example 8: Solve $\frac{n}{-2} = 7$

Multiply both sides by $-2$:

$\frac{n}{-2} \times (-2) = 7 \times (-2)$

$n = -14$

Check: $\frac{-14}{-2} = 7$ — confirmed.

Why Checking Matters

Always substitute your answer back into the original equation to verify. This takes only a few seconds and catches arithmetic errors before they become bigger problems — especially important in applied settings where an incorrect answer could mean a wrong dosage or a financial error.

Equations with Fractions and Decimals

One-step equations can involve fractions and decimals too. The same inverse-operation approach applies.

Example 9: Solve $x + 2.5 = 6.8$

$x = 6.8 - 2.5 = 4.3$

Check: $4.3 + 2.5 = 6.8$ — correct.

Example 10: Solve $\frac{1}{3}n = 12$

The variable is multiplied by $\frac{1}{3}$. Multiply both sides by 3 (the reciprocal of $\frac{1}{3}$):

$3 \cdot \frac{1}{3}n = 3 \cdot 12$

$n = 36$

Check: $\frac{1}{3}(36) = 12$ — confirmed.

Real-World Application: Nursing — Finding a Missing Dose

A nurse knows that after administering a dose, the patient’s total medication for the day is 450 mg. The patient already received 200 mg in the morning. How much was the afternoon dose?

Let $d$ represent the afternoon dose:

$200 + d = 450$

Subtract 200 from both sides:

$d = 450 - 200 = 250$

Answer: The afternoon dose was 250 mg. The nurse can verify: $200 + 250 = 450$ mg total — that checks out.

Real-World Application: Retail — Unit Pricing

A retail associate sold 6 identical items and the total sale was $54. What was the price per item?

Let $p$ represent the price per item:

$6p = 54$

Divide both sides by 6:

$p = \frac{54}{6} = 9$

Answer: Each item costs $9. Check: $6 \times 9 = 54$ — confirmed.

Common Mistakes to Avoid

1. Using the wrong inverse operation. If the equation is $x + 5 = 12$, you need subtraction (not division). Match the inverse to the operation that is applied to the variable.

2. Operating on only one side. Whatever you do to the left side, you MUST do to the right side. If you subtract 5 from the left, subtract 5 from the right.

3. Sign errors with negatives. In $x - 8 = -3$, adding 8 to both sides gives $x = -3 + 8 = 5$, not $x = -11$. Pay close attention when adding to a negative number.

4. Forgetting to check. Substituting your answer back into the original equation takes seconds and catches mistakes. Make it a habit.

5. Confusing the variable side. If the equation is $15 = x + 7$, you still subtract 7 from both sides: $15 - 7 = x$, so $x = 8$. The variable does not have to be on the left.

Practice Problems

Test your understanding with these problems. Click to reveal each answer.

Key Takeaways

• An equation states that two expressions are equal; solving means finding the variable’s value
• Think of the equation as a balance scale — whatever you do to one side, do to the other
• Use the inverse operation to undo what is happening to the variable: addition undoes subtraction, multiplication undoes division (and vice versa)
• Always check your solution by substituting it back into the original equation
• One-step equations form the foundation for multi-step equation solving — master them now, and two-step equations will feel natural

Return to Pre-Algebra for more topics in this section.