Lesson 2: Solve Equations Using the Division and Multiplication Properties of Equality

New Concept

This lesson introduces two powerful tools: the Division and Multiplication Properties of Equality. You'll learn to use these inverse operations to isolate a variable and solve for its value, even in complex, multi-step equations.

What’s next

Now, let's see these properties in action. You'll work through interactive examples and then apply your skills on a series of practice cards.

Property

Division Property of Equality: For all real 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 the same non-zero quantity, you still have equality. The goal is to 'undo' the operation on the variable.

Examples

• To solve $4x = -28$, we 'undo' the multiplication by dividing both sides by 4. This gives $\frac{4x}{4} = \frac{-28}{4}$, so $x = -7$.

Property

Multiplication Property of Equality: For all real numbers $a$, $b$, $c$, if $a = b$, then $ac = bc$.

When you multiply both sides of an equation by the same quantity, you still have equality. This is used to 'undo' division in an equation.

Examples

• To solve $\frac{a}{-7} = -42$, we multiply both sides by $-7$. This results in $-7(\frac{a}{-7}) = -7(-42)$, which simplifies to $a = 294$.

Property

Since the product of a number and its reciprocal is 1, our strategy will be to isolate the variable by multiplying by the reciprocal of the fractional coefficient. For an equation like $\frac{a}{b}x = c$, you multiply by $\frac{b}{a}$.

Examples

• To solve $\frac{2}{3}x = 18$, multiply both sides by the reciprocal of $\frac{2}{3}$, which is $\frac{3}{2}$. This gives $\frac{3}{2} \cdot \frac{2}{3}x = \frac{3}{2} \cdot 18$, so $x = 27$.

• To solve $\frac{3}{4}r = 15$, multiply both sides by $\frac{4}{3}$. You get $\frac{4}{3} \cdot \frac{3}{4}r = \frac{4}{3} \cdot 15$, which simplifies to $r = 20$.

Property

Many equations start out more complicated than they need to be. Before applying the properties of equality, simplify both sides of the equation as much as possible by combining like terms or using the distributive property.

Examples

• To solve $8x + 9x - 5x = -3 + 15$, first combine like terms on each side to get $12x = 12$. Then, divide by 12 to find $x = 1$.

• In $-3(n - 2) - 6 = 21$, first distribute the $-3$ to get $-3n + 6 - 6 = 21$. This simplifies to $-3n = 21$. Dividing by $-3$ gives $n = -7$.