Lesson 4: Write and Solve Multiplication and Division Equations

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 $a$, $b$, and $c$, and $c \neq 0$, if $a = b$, then $\frac{a}{c} = \frac{b}{c}$. When you divide both sides of an equation by any non-zero number, you still have equality. The goal in solving an equation is to 'undo' the operation on the variable. If a variable is multiplied by a number, we divide both sides by that number to 'undo' the multiplication.

Examples

• To solve $9x = 72$, we divide both sides by 9. This gives $\frac{9x}{9} = \frac{72}{9}$, which simplifies to $x = 8$.
• To solve $6y = 48$, we divide both sides by $6$. This gives $\frac{6y}{6} = \frac{48}{6}$, which simplifies to $y = 8$.
• To solve $5p = 15$, we divide both sides by 5. This gives $\frac{5p}{5} = \frac{15}{5}$, which simplifies to $p = 3$.

Explanation

Think of this as fair sharing. If two sides of an equation are balanced, dividing both by the same non-zero amount keeps them balanced. This is how we isolate a variable that's being multiplied by a number.

Property

When solving real-world problems that involve multiplication or division equations, use the Division Property of Equality (divide both sides by the same number) or the Multiplication Property of Equality (multiply both sides by the same number) to isolate the variable and find the solution.

Examples