Lesson 5: Solve Equations Using Integers; The Division Property of Equality

New Concept

Expand your equation-solving toolkit by learning the Division Property of Equality. You'll learn how to isolate a variable by "undoing" multiplication, allowing you to solve equations like $7x = -49$ and translate word problems into algebra.

What’s next

Get ready to see this property in action! You’ll tackle interactive examples and then apply your new skills in a series of practice cards.

Property

How to determine whether a number is a solution to an equation.

Step 1. Substitute the number for the variable in the equation.
Step 2. Simplify the expressions on both sides of the equation.
Step 3. Determine whether the resulting equation is true.

• If it is true, the number is a solution.
• If it is not true, the number is not a solution.

Examples

• Is $x=3$ a solution to $4x - 7 = 5$? We check: $4(3) - 7 = 12 - 7 = 5$. Since $5=5$, yes, it is a solution.

Property

When you add or subtract the same quantity from both sides of an equation, you still have equality.

Subtraction Property of Equality
For any numbers $a$, $b$, $c$, if $a = b$ then $a - c = b - c$.

Addition Property of Equality
For any numbers $a$, $b$, $c$, if $a = b$ then $a + c = b + c$.

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 $y = -8$.

Property

To solve word problems, first translate the word sentences into equations with variables, and then solve the equations using the properties of equality. Identify keywords to determine the correct operation.

Examples

• 'Seven more than $n$ is equal to 2.' translates to $n + 7 = 2$. Subtracting 7 from both sides gives $n = -5$.

• 'The difference of $p$ and 9 is $-15$.' translates to $p - 9 = -15$. Adding 9 to both sides gives $p = -6$.