Lesson 5: Solving Two-Step Equations

Property

Addition Property of Equality: Adding the same number to both sides of an equation maintains the equality. If $a=b$, then $a+c = b+c$.
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 $ac = bc$. These properties are used to isolate variables.

Examples

• Given $x - 9 = 12$, we use the Addition Property of Equality to add 9 to both sides: $x - 9 + 9 = 12 + 9$, which gives $x = 21$.
• Given $6y = 42$, we use the Multiplication Property of Equality to divide both sides by 6: $\frac{6y}{6} = \frac{42}{6}$, which gives $y = 7$.
• Given $\frac{z}{5} = 3$, we use the Multiplication Property of Equality to multiply both sides by 5: $\frac{z}{5} \cdot 5 = 3 \cdot 5$, which gives $z = 15$.

Explanation

An equation is like a balanced scale. To keep it perfectly balanced, whatever operation you perform on one side (like adding 5), you must do the exact same thing to the other side. This ensures the equation remains true.

Property

1. Add or subtract the same number to both sides of the equation.

2. Multiply or divide both sides of an equation by the same nonzero number.

Property

To solve real-world problems involving two operations, work backwards from the final result by reversing each operation in opposite order: if the original operations were "multiply then add," work backwards by "subtract then divide."

Examples