Skip to content
mathlab
0
0
0 XP
Lvl 1
KJ
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.
0/5QUESTS

Printable Worksheets

Print or save as PDF, or build a custom worksheet from any module's questions.

Build Apply Master Build custom
Think First
warm-up

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

Record in workbook.
1
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 $\frac{x}{a} = b$
2
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 $x = 15$.
3
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 ÷ 3 on both sides x = 4
$3x = 12 \Rightarrow x = 4$
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
$-4x = 20 \Rightarrow \frac{-4x}{-4} = \frac{20}{-4} \Rightarrow x = -5$
4
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 $x = ab$.

x/3 = 5 × 3 on both sides x = 15
$\frac{x}{3} = 5 \Rightarrow x = 15$
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
$\frac{x}{-2} = 6 \Rightarrow (-2) \times \frac{x}{-2} = (-2) \times 6 \Rightarrow x = -12$
5
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
+15 XP per step
Q
PROBLEM
Solve: (a) $6x = 42$    (b) $\frac{x}{5} = 8$
  1. 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. 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. 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. 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: $40 \div 5 = 8$ ✓
6
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

1
Solve $5x = 35$.
+10 XP
2
Solve $\frac{x}{4} = 7$.
+10 XP
3
Solve $-3x = 24$.
+10 XP
4
Solve $\frac{x}{-5} = -3$.
+10 XP
5
"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

Extension Problems

Ready for a bigger challenge? Try these extension problems.

R
Quick Review

Key Concept

Review the main ideas from this lesson.

Formulas

Key formulas and rules.

Watch Out

Common mistakes to avoid.

Check

Always verify your answers.

Practice

Keep practicing to master.

Next

Build on these skills.

Interactive: Algebra Machine

Substitute numbers into algebraic expressions and see them evaluate step by step.

Your Badges

0 of 6
First Steps
3-Day Streak
3 in a Row
Lesson Ace
Stretch Seeker
Daily Warrior
DAILY CHALLENGE · TODAY
The Algebra Sprint
Solve 5 algebra problems in under 60 seconds for double coins!
NEXT UP · 0% UNLOCKED
Next Lesson
0/5
Finish this lesson's Practice MCs to unlock!

Mark lesson as complete

Tick when you've finished Learn, Practice and the Stretch. Earns +90 XP and +25 coins.

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.

$3x = 27$
$\frac{x}{2} = 8$
$6x = -42$
$\frac{x}{-4} = 5$
$5x = 0$
$\frac{x}{10} = -3$
$-2x = 18$
$\frac{2x}{5} = 4$
$9x = -72$
$\frac{x}{-3} = -6$
Want help with Lesson 13: Solving One-Step Equations (Multiply/Divide)?

Work through this topic 1-on-1 with an experienced maths tutor.

Book a free session →