Lesson 21: Solving One-Step Equations by Multiplying or Dividing

New Concept

To find the solution of an equation, isolate the variable by using inverse operations. You must use the same inverse operation on each side of the equation.

Multiplication Property of Equality
If $a = b$, then $ac = bc$.

Division Property of Equality
If $a = b$ and $c \neq 0$, then $\frac{a}{c} = \frac{b}{c}$.

What’s next

This is just the start. Soon, you'll walk through worked examples and apply this skill to solve a real-world architecture problem.

Property

An inverse operation undoes another operation.
Add ↔ Subtract
Multiply ↔ Divide

Examples

To solve $\frac{x}{5} = 10$, you undo the division by multiplying both sides by 5, so $x = 50$.
To solve $3x = 15$, you undo the multiplication by dividing both sides by 3, so $x = 5$.
To solve $\frac{2}{3}y = 8$, you undo multiplication by $\frac{2}{3}$ by multiplying by its inverse, $\frac{3}{2}$, so $x = 16$

Explanation

Think of these as your equation-solving superpowers! To get a variable alone, just use the opposite action. If a variable is being divided, you multiply to set it free. If it's being multiplied, you divide. It's like hitting the rewind button on math to find the hidden answer!

Property

Both sides of an equation can be multiplied by the same number, and the statement will still be true.
$a=b$, so $ac=bc$

Examples

To solve $\frac{x}{6} = 8$, multiply both sides by 6: $6 \cdot \frac{x}{6} = 8 \cdot 6$, which simplifies to $x = 48$.
For $-11 = \frac{1}{4}w$, multiply both sides by 4: $4 \cdot -11 = 4 \cdot \frac{1}{4}w$, which gives you $-44 = w$.

Explanation

Imagine an equation is a balanced scale. If you add weight to one side, you have to add the same weight to the other to keep it level. The Multiplication Property is the same idea! Multiplying both sides by the same number keeps your equation perfectly balanced and true, letting you solve for the variable.

Property

Both sides of an equation can be divided by the same number, and the statement will still be true.
If $a=b$, then $\frac{a}{c} = \frac{b}{c}$ (where $c \ne 0$).

Examples

To solve $5x = 20$, divide both sides by 5: $\frac{5x}{5} = \frac{20}{5}$, which simplifies to $x = 4$.
In the equation $-12 = 3n$, divide both sides by 3: $\frac{-12}{3} = \frac{3n}{3}$, which gives you $-4 = n$.

Explanation

Just like with multiplying, you have to keep things fair and balanced! If you divide one side of your equation by a number, you must do the exact same thing to the other side. This ensures your equation stays equal, helping you march steadily toward finding the value of your hidden variable.