Lesson 3: Solving Equations Using Multiplication or Division

Property

Multiplication and division are inverse operations, which means they "undo" each other. For any number $x$ and any non-zero number $a$:

Examples

Property

For any numbers $a$, $b$, and $c$, if $a = b$, then $ac = bc$.
If you multiply both sides of an equation by the same number, you still have equality.
To 'undo' division in an equation, we multiply both sides by the number the variable is divided by.

Examples

• To solve $\frac{x}{7} = 5$, we multiply both sides by 7. This gives $7 \cdot \frac{x}{7} = 7 \cdot 5$, which simplifies to $x = 35$.
• To solve $\frac{m}{4} = 11$, we multiply both sides by $4$. This gives $4 \cdot \frac{m}{4} = 4 \cdot 11$, which simplifies to $m = 44$.
• To solve $n \div 3 = 6$, we multiply both sides by $3$. This gives $n \div 3 \times 3 = 6 \times 3$, so $n = 18$.

Explanation

If two quantities are perfectly equal, multiplying both by the same amount won't change their equality. We use this trick to cancel out division and solve for a variable that is part of a fraction.

Property

For any numbers $a$, $b$, $c$, and $c \neq 0$, if $a = b$ then $\frac{a}{c} = \frac{b}{c}$.
When you divide both sides of an equation by any nonzero number, you still have equality.
This is used to solve equations of the form $ax=b$ by isolating the variable.

Examples

• To solve $6x = -54$, we divide both sides by 6. This gives $\frac{6x}{6} = \frac{-54}{6}$, so $x = -9$.

• To solve $-4y = 32$, we divide both sides by $-4$. This gives $\frac{-4y}{-4} = \frac{32}{-4}$, so $y = -8$.