Lesson 13~30 minUnit 2 · Patterns & Algebra+90 XP

Solving One-Step Equations (Multiply/Divide)

Undo multiplication by dividing and division by multiplying, including equations where the coefficient is negative.

Today’s hook: $-3x = 12$ catches people out. Dividing both sides by $-3$ gives $x = -4$, not $4$. The sign travels with the number you divide by, every time.

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Think First

warm-up

Before diving into the practice questions, take a moment to think about what you already know about this topic.

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What You'll Master

objectives

Know

The inverse of multiplication is division (and vice versa)

Understand

To isolate $x$, divide or multiply both sides by the same value

Can Do

Solve equations of the form $ax = b$
Solve equations of the form $\frac{x}{a} = b$

Words You Need

vocabulary

Multiplicative InverseThe number you multiply by to get 1. For 3, the multiplicative inverse is $\frac{1}{3}$, because $3 \times \frac{1}{3} = 1$. Dividing by 3 is the same as multiplying by $\frac{1}{3}$.

CoefficientThe number multiplying the variable in a term. In $7x$, the coefficient is 7. In $-4x$, the coefficient is $-4$.

IsolateTo get the variable by itself on one side of the equals sign. The goal when solving any equation.

Divide Both SidesThe operation used to undo multiplication. If $3x = 12$, divide both sides by 3 to get $x = 4$.

Multiply Both SidesThe operation used to undo division. If $\frac{x}{3} = 5$, multiply both sides by 3 to get $x = 15$.

Undoing Multiplication

+5 XP

If $x$ is multiplied by a number, divide both sides by that number to isolate $x$. The number multiplying $x$ is called the coefficient.

The key idea: multiplication and division are inverse operations they undo each other. Dividing by the coefficient cancels it out, leaving $x$ alone on one side.

$3x = 12 \Rightarrow x = 4$

Example 1

$3x = 12 \Rightarrow \frac{3x}{3} = \frac{12}{3} \Rightarrow x = 4$

Example 2

$7x = 35 \Rightarrow \frac{7x}{7} = \frac{35}{7} \Rightarrow x = 5$

Example 3

$-4x = 20 \Rightarrow \frac{-4x}{-4} = \frac{20}{-4} \Rightarrow x = -5$

Undoing Division

+5 XP

If $x$ is divided by a number, multiply both sides by that number to isolate $x$. Multiplying by the divisor cancels the division.

When you see $\frac{x}{a} = b$, the $x$ is being divided by $a$. To undo this, multiply both sides by $a$. The $a$'s cancel on the left, leaving $x = ab$.

$\frac{x}{3} = 5 \Rightarrow x = 15$

Example 1

$\frac{x}{3} = 5 \Rightarrow 3 \times \frac{x}{3} = 3 \times 5 \Rightarrow x = 15$

Example 2

$\frac{x}{4} = 7 \Rightarrow 4 \times \frac{x}{4} = 4 \times 7 \Rightarrow x = 28$

Example 3

$\frac{x}{-2} = 6 \Rightarrow (-2) \times \frac{x}{-2} = (-2) \times 6 \Rightarrow x = -12$

Negative Coefficients

+5 XP

Key insight: Dividing both sides by a negative number flips the sign of the answer. Use the sign rules carefully.

Positive ÷ Negative

$-5x = 25 \Rightarrow x = \frac{25}{-5} = -5$ (positive divided by negative = negative)

Negative ÷ Negative

$-2x = -14 \Rightarrow x = \frac{-14}{-2} = 7$ (negative divided by negative = positive)

Memory tip

Same signs → positive answer. Different signs → negative answer.

Quick check: Does $-3x = -12$ give a positive or negative answer? Answer: $x = \frac{-12}{-3} = 4$, positive! Negative ÷ negative = positive.

Watch Me Solve It · Worked examples

Watch Me Solve It · Two equation types

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PROBLEM

Solve: (a) $6x = 42$ (b) $\frac{x}{5} = 8$

1

Identify the operation on $x$ for part (a)

$6x = 42$, $x$ is multiplied by 6

The coefficient is 6. Multiplication needs to be undone by dividing.
2

Divide both sides by 6

$\frac{6x}{6} = \frac{42}{6} \Rightarrow x = 7$

Dividing both sides by the same number keeps the equation balanced. Check: $6 \times 7 = 42$ ✓
3

Identify the operation on $x$ for part (b)

$\frac{x}{5} = 8$, $x$ is divided by 5

Division needs to be undone by multiplying both sides by 5.
4

Multiply both sides by 5

$5 \times \frac{x}{5} = 5 \times 8 \Rightarrow x = 40$

Multiplying undoes the division. Check: $40 \div 5 = 8$ ✓

Answers (a) $x = 7$ · check: $6 \times 7 = 42$ ✓ (b) $x = 40$ · check: $40 \div 5 = 8$ ✓

Common Pitfalls

heads-up

Multiplying instead of dividing to undo multiplication

If $5x = 35$, a student mistakenly multiplies by 5, getting $x = 175$. Wrong! Multiplication is undone by division.

Fix: always ask "what is being done to $x$?" then do the opposite. $5x = 35 \Rightarrow x = 35 \div 5 = 7$.

Dividing instead of multiplying to undo division

If $\frac{x}{3} = 9$, a student mistakenly divides by 3, getting $x = 3$. Wrong! Division is undone by multiplication.

Fix: $\frac{x}{3} = 9 \Rightarrow x = 9 \times 3 = 27$. Multiply both sides by the divisor.

Assuming a negative coefficient always gives a negative answer

$-3x = -12$ gives $x = 4$, which is positive! The sign of the answer depends on both the coefficient and the right-hand side.

Fix: always divide and apply sign rules. Negative ÷ negative = positive, so $x = \frac{-12}{-3} = 4$.

Quick Check · 5 questions

Solve $5x = 35$.

+10 XP

Solve $\frac{x}{4} = 7$.

+10 XP

Solve $-3x = 24$.

+10 XP

Solve $\frac{x}{-5} = -3$.

+10 XP

"A number multiplied by 6 equals 42." Which equation and solution are correct?

+10 XP

Apply Easy 2 MARKS

Q1. Solve $8x = -64$. Show your working and check your answer.

Answer in your workbook.

Apply Medium 3 MARKS

Q2. Solve $\frac{x}{7} = -5$. Show your working and check your answer.

Answer in your workbook.

Apply Hard 4 MARKS

Q3. A rectangle has an area of $72 \text{ cm}^2$ and a width of $8 \text{ cm}$. Write and solve an equation to find the length.

Answer in your workbook.

Stretch Challenge · +25 XP, +10 coins

Choose the correct inverse operation

Choose the correct inverse operation. Compare the equations $x\div4=3$ and $4y=12$. A student says both are solved by dividing the right-hand side by 4.

Solve each equation using its inverse operation, then check by substitution. Explain why the student’s rule works for only one of these equations.

Show your working first, then open the worked solution to check your reasoning.

Check the worked solution

For $x\div4=3$, multiply both sides by 4 to get $x=12$; check $12\div4=3$. For $4y=12$, divide both sides by 4 to get $y=3$; check $4(3)=12$. Division undoes multiplication in the second equation. Multiplication is needed to undo division in the first; the inverse depends on the operation applied to the unknown.

Quick Review

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Formulas

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Watch Out

Common mistakes to avoid.

Check

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← Solving One-Step Equations (Add/Subtract) Solving Two-Step Equations →

Unit overview · Maths subject page · Unit quiz

Brain Trainer

⚡

Speed Drills, Solve These Equations!

Set a timer for 3 minutes. Solve as many as you can. Check your answers after.

$3x = 27$

$\frac{x}{2} = 8$

$6x = -42$

$\frac{x}{-4} = 5$

$5x = 0$

$\frac{x}{10} = -3$

$-2x = 18$

$\frac{2x}{5} = 4$

$9x = -72$

$\frac{x}{-3} = -6$

Solving One-Step Equations (Multiply/Divide)

Printable Worksheets

Know

Understand

Can Do

Choose the correct inverse operation

Key Concept

Formulas

Watch Out

Check

Practice

Next

Interactive: Algebra Machine

Your Badges

Mark lesson as complete

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