Lesson 38: Using Properties of Equality to Solve Equations

New Concept

Welcome to Saxon Math! This course introduces algebra, where we use variables and equations to model the world and find unknown values with logic.

What’s next

To begin, we'll explore the fundamental 'rules of the game': the properties of equality. You'll then apply these rules using inverse operations in worked examples.

Property

Inverse operations are operations that 'undo' each other. Addition and subtraction are inverses ($n + 5 - 5 = n$), and multiplication and division are inverses ($n \times 5 \div 5 = n$).

Examples

To solve $x + 8 = 15$, use the inverse of addition. Subtract 8 from both sides: $x + 8 - 8 = 15 - 8$, so $x = 7$.
To solve $3m = 21$, use the inverse of multiplication. Divide both sides by 3: $\frac{3m}{3} = \frac{21}{3}$, so $m = 7$.

Explanation

Think of an equation as a balance scale. To find the unknown 'x', you undo whatever operation is with it. Just do the same to both sides to keep it perfectly balanced and find your answer! This process is called isolating the variable, which means getting it all alone on one side.

Property

If $a = b$, you can perform the same operation on both sides and the equation remains true.
Addition: $a + c = b + c$.
Subtraction: $a - c = b - c$.
Multiplication: $ac = bc$.
Division: $\frac{a}{c} = \frac{b}{c}$ (as long as $c$ is not 0).

Examples

Given $y - 6 = 11$, use the Addition Property: $y - 6 + 6 = 11 + 6$, so $y = 17$.
Given $4p = 24$, use the Division Property: $\frac{4p}{4} = \frac{24}{4}$, so $p = 6$.
A pack of 5 markers costs 10 dollars. To find the cost per marker ($c$), solve $5c = 10$. Use Division: $\frac{5c}{5} = \frac{10}{5}$, so $c = 2$ dollars.

Explanation

Imagine an equation is a perfectly balanced seesaw. If you add a brick to one side, you must add an identical brick to the other side to keep it from tipping over! This simple rule lets you manipulate equations to solve for the variable while keeping everything fair and equal.