Solving One-Step Equations — Examples and Practice

Grade: 6-7 | Topic: Algebra

What You Will Learn

After this lesson you will be able to solve any one-step equation confidently. You will understand what inverse operations are, know which operation to apply for each equation type, and be able to check your answer every time. These skills form the foundation for all equation-solving in algebra.

Theory

What is a one-step equation?

A one-step equation is an equation that can be solved in a single step. Only one operation (addition, subtraction, multiplication, or division) separates the variable from its value, so you only need one inverse operation to isolate it.

Here are some typical one-step equations:

Equation	What is happening to $x$	Inverse operation
$x + 9 = 14$	9 is added	Subtract 9
$x - 3 = 10$	3 is subtracted	Add 3
$4x = 28$	$x$ is multiplied by 4	Divide by 4
$\dfrac{x}{5} = 6$	$x$ is divided by 5	Multiply by 5

Inverse operations

An inverse operation undoes what another operation does. Think of it like pressing "undo" on a calculator:

$\text{Addition} \longleftrightarrow \text{Subtraction}$

$\text{Multiplication} \longleftrightarrow \text{Division}$

The key principle is the balance rule: whatever you do to one side of the equation, you must do exactly the same thing to the other side.

$\text{If } a = b, \text{ then } a + c = b + c$

$\text{If } a = b, \text{ then } a \times c = b \times c \quad (c \neq 0)$

The solving process

For every one-step equation, follow this pattern:

1. Identify the operation being performed on the variable.
2. Apply the inverse operation to both sides of the equation.
3. Simplify to find the value of the variable.
4. Check by substituting the answer back into the original equation.

Worked Examples

Example 1: Addition equation (easy)

Problem: Solve $x + 8 = 15$

Step 1: The variable $x$ has 8 added to it, so subtract 8 from both sides.

$x + 8 - 8 = 15 - 8$

$x = 7$

Answer: $x = 7$

Check: $7 + 8 = 15$ ✓

Example 2: Subtraction equation (easy)

Problem: Solve $x - 12 = 5$

Step 1: The variable $x$ has 12 subtracted from it, so add 12 to both sides.

$x - 12 + 12 = 5 + 12$

$x = 17$

Answer: $x = 17$

Check: $17 - 12 = 5$ ✓

Example 3: Multiplication equation (medium)

Problem: Solve $6x = 54$

Step 1: The variable $x$ is multiplied by 6, so divide both sides by 6.

$\frac{6x}{6} = \frac{54}{6}$

$x = 9$

Answer: $x = 9$

Check: $6 \times 9 = 54$ ✓

Example 4: Division equation (medium)

Problem: Solve $\dfrac{x}{4} = 9$

Step 1: The variable $x$ is divided by 4, so multiply both sides by 4.

$\frac{x}{4} \times 4 = 9 \times 4$

$x = 36$

Answer: $x = 36$

Check: $\frac{36}{4} = 9$ ✓

Example 5: Equation with a negative result (medium)

Problem: Solve $x + 15 = 7$

Step 1: Subtract 15 from both sides.

$x + 15 - 15 = 7 - 15$

$x = -8$

Answer: $x = -8$

Check: $-8 + 15 = 7$ ✓

This example shows that solutions can be negative numbers. That is perfectly valid.

Common Mistakes

Mistake 1: Using the wrong inverse operation

❌ $x + 5 = 12 \implies x = 12 + 5 = 17$ (added instead of subtracting)

✅ $x + 5 = 12 \implies x = 12 - 5 = 7$

Why this matters: Addition is undone by subtraction, not more addition. If the equation adds a number to $x$, you need to subtract that same number from both sides to cancel it out.

Mistake 2: Applying the operation to only one side

❌ $x - 3 = 10 \implies x = 10$ (added 3 to the left but not the right)

✅ $x - 3 = 10 \implies x - 3 + 3 = 10 + 3 \implies x = 13$

Why this matters: An equation is only true when both sides are equal. Changing one side without changing the other destroys that equality and gives a wrong answer.

Mistake 3: Confusing multiplication and division signs

❌ $\frac{x}{3} = 7 \implies x = \frac{7}{3}$ (divided again instead of multiplying)

✅ $\frac{x}{3} = 7 \implies x = 7 \times 3 = 21$

Why this matters: When $x$ is being divided by a number, you undo that by multiplying, not dividing further. The inverse of $\div 3$ is $\times 3$.

Practice Problems

Try these on your own before checking the answers:

1. $x + 11 = 20$
2. $x - 7 = 15$
3. $5x = 45$
4. $\dfrac{x}{8} = 3$
5. $x + 25 = 13$

Click to see answers

1. $x = 9$ — Subtract 11 from both sides: $20 - 11 = 9$.
2. $x = 22$ — Add 7 to both sides: $15 + 7 = 22$.
3. $x = 9$ — Divide both sides by 5: $45 \div 5 = 9$.
4. $x = 24$ — Multiply both sides by 8: $3 \times 8 = 24$.
5. $x = -12$ — Subtract 25 from both sides: $13 - 25 = -12$.

Summary

• A one-step equation needs exactly one inverse operation to solve.
• Inverse operations undo each other: addition/subtraction and multiplication/division are inverse pairs.
• Always perform the same operation on both sides of the equation to keep it balanced.
• Check your answer by substituting it back into the original equation.

Related Topics

• Linear Equations — How to Solve Step by Step
• Solving Two-Step Equations Step by Step
• Integers — Operations, Number Line, and Word Problems

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