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Learn intro · examples
Practice 5 MC + 3 short answer
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Before diving into the practice questions, take a moment to think about what you already know about this topic.
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 Inverse The 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 $\frac{1}{3}$.
Coefficient The number multiplying the variable in a term. In $7x$, the coefficient is 7. In $-4x$, the coefficient is $-4$.
Isolate To get the variable by itself on one side of the equals sign. The goal when solving any equation.
Divide Both Sides The operation used to undo multiplication. If $3x = 12$, divide both sides by 3 to get $x = 4$.
Multiply Both Sides The operation used to undo division. If $\frac{x}{3} = 5$, multiply both sides by 3 to get $x = 15$.
3
Undoing Multiplication
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
$-4x = 20 \Rightarrow \frac{-4x}{-4} = \frac{20}{-4} \Rightarrow x = -5$
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
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
Q
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: $40 \div 5 = 8$ ✓
Show me the next step
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 $5x = 35$.
A OPTION A $x = 175$ B OPTION B $x = 7$ C OPTION C $x = 30$ D OPTION D $x = 40$
2
Solve $\frac{x}{4} = 7$.
A OPTION A $x = \frac{7}{4}$ B OPTION B $x = 11$ C OPTION C $x = 28$ D OPTION D $x = 3$
3
Solve $-3x = 24$.
A OPTION A $x = 21$ B OPTION B $x = -8$ C OPTION C $x = 8$ D OPTION D $x = -21$
4
Solve $\frac{x}{-5} = -3$.
A OPTION A $x = -\frac{3}{5}$ B OPTION B $x = \frac{3}{5}$ C OPTION C $x = -15$ D OPTION D $x = 15$
5
"A number multiplied by 6 equals 42." Which equation and solution are correct?
A OPTION A $6x = 42$, $x = 7$ B OPTION B $x + 6 = 42$, $x = 36$ C OPTION C $6x = 42$, $x = 36$ D OPTION D $\frac{x}{6} = 42$, $x = 7$
Apply
Easy
2 MARKS
Q1. Solve $8x = -64$. Show your working and check your answer.
Apply
Medium
3 MARKS
Q2. Solve $\frac{x}{7} = -5$. Show your working and check your answer.
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.
Extension Problems
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Unit overview · Maths subject page · Unit quiz
⚡
Speed Drills — Solve These Equations!
Set a timer for 3 minutes. Solve as many as you can. Check your answers after.
$3x = 27$ ?
$\frac{x}{2} = 8$ ?
$6x = -42$ ?
$\frac{x}{-4} = 5$ ?
$5x = 0$ ?
$\frac{x}{10} = -3$ ?
$-2x = 18$ ?
$\frac{2x}{5} = 4$ ?
$9x = -72$ ?
$\frac{x}{-3} = -6$ ?