Lesson 41: Using Formulas, Distributive Property

New Concept

The Distributive Property lets you multiply a sum by 'spreading' the multiplication over the terms being added or subtracted within parentheses.

The Distributive Property is often expressed in equation form using variables:

The Distributive Property also applies over subtraction:

What’s next

Next, you'll see worked examples that use the Distributive Property to simplify expressions and connect two different formulas for a rectangle's perimeter.

Property

To use a formula like $A = lw$, replace the letters (variables) with their given number values and then perform the calculation to find the final answer.

Examples

Find area $A = lw$ when $l$ is 8 ft and $w$ is 4 ft: $A = (8 \text{ ft})(4 \text{ ft}) = 32 \text{ ft}^2$.
Evaluate $2(l+w)$ when $l=8$ cm and $w=4$ cm: $2(8 \text{ cm} + 4 \text{ cm}) = 2(12 \text{ cm}) = 24 \text{ cm}$.

Explanation

Think of formulas as secret math recipes. You just substitute the letter ingredients with their given numbers, follow the cooking instructions (add, multiply, etc.), and you'll bake up the correct answer every single time! It’s your trusty guide to solving problems.

Property

$a(b + c) = ab + ac$
$a(b - c) = ab - ac$

Examples

Show that $2(l+w)$ and $2l+2w$ are equal for $l=30, w=20$: $2(30+20)=2(50)=100$ and $2(30)+2(20)=60+40=100$.
Simplify using the property: $2(n+5) = 2 \cdot n + 2 \cdot 5 = 2n+10$.

Explanation

This property is your ticket to breaking open parentheses! The term outside gets "distributed" or multiplied by every single term inside. It's like a pizza delivery—the number outside brings a slice of multiplication to everyone waiting inside the parentheses. No one gets left out!