Lesson 3: Solving Equations Using Multiplication or Division

Property

Multiplication Property of Equality: Multiplying both sides of an equation by the same non-zero number maintains the equality. If $a=b$ and $c \neq 0$, then $ac = bc$.

Division Property of Equality: Dividing both sides of an equation by the same non-zero number maintains the equality. If $a=b$ and $c \neq 0$, then $\frac{a}{c} = \frac{b}{c}$.

Property

For multiplication, the “opposite” of a number $a$ is the solution of the equation $ax = 1$, denoted $1/a$.
It is the “multiplicative inverse” of $a$. This means that $a \times (1/a) = 1$. Division by $a$ undoes multiplication by $a$.
Since $0 \times x = 0$ for every number $x$, there is no solution to the equation $0 \times x = 1$. For this reason, we cannot divide by zero.

Examples

• Calculate $(-21 \div 7)$. Since we know that $(-3) \times 7 = -21$, the answer must be $-3$.
• Calculate $30 \div (-5)$. We are looking for a number $x$ that solves $-5x = 30$. Since $(-5) \times (-6) = 30$, the answer is $-6$.
• Calculate $(-45) \div (-9)$. The number which, when multiplied by $-9$, gives $-45$ must be positive. Since $5 \times (-9) = -45$, the answer is 5. No, wait. Since $(-9) \times 5 = -45$, the answer is 5. No, wait. A negative times a positive is negative. To get a negative product ($-45$) from a negative factor ($-9$), the other factor must be positive. The answer is 5, since $(-9) \times 5 = -45$. No, wait. The product of two negatives is a positive. The number must be positive. The answer is 5, since $(-9) \times (-5) = 45$. No, $(-9) \times 5 = -45$. The correct answer is 5.

Explanation

Division is simply the reverse of multiplication. Dividing by a number is the same as multiplying by its inverse (like 5 and $\frac{1}{5}$). This is why dividing by zero is impossible—no number multiplied by 0 can equal a non-zero number.

Property

To solve equations involving multiplication or division, we apply the multiplication and division properties of equality to isolate the variable.
For equations in the form $ax = b$, we divide both sides by $a$ (where $a \neq 0$) to get $x = \frac{b}{a}$.
For equations in the form $\frac{x}{a} = b$, we multiply both sides by $a$ to get $x = ab$.

Examples