Lesson 1: The Distributive Law

New Concept

Master the distributive law, $a(b+c)=ab+ac$, to simplify expressions containing parentheses. This skill is essential for solving more complex linear equations and writing algebraic models for applied problems.

What’s next

Next, you’ll work through interactive examples and practice cards to apply this law when solving equations and tackling real-world problems.

Property

To simplify a product such as $3(2a)$, we multiply the numerical factors to find $3(2a) = (3 \cdot 2)a = 6a$. This calculation is an application of the associative law for multiplication. We can see why the product is reasonable by recalling that multiplication is repeated addition, so that $3(2a) = 2a + 2a + 2a = 6a$.

Examples

• To simplify $4(5b)$, you multiply the numerical factors: $(4 \cdot 5)b = 20b$.
• To simplify $-6(2y)$, you multiply the numbers: $(-6 \cdot 2)y = -12y$.
• To simplify $-3(-8z)$, you multiply the coefficients: $(-3 \cdot -8)z = 24z$.

Explanation

When a number is outside parentheses with a single term inside, just multiply the numbers together. Think of it as a shortcut for adding the term multiple times. It's all about grouping the numbers first to simplify.

Property

If $a$, $b$, and $c$ are any numbers, then

If the terms inside parentheses are not like terms, we have no choice but to use the distributive law to simplify the expression.

Examples

• To simplify $5(x+4)$, distribute the 5 to each term: $5(x) + 5(4) = 5x + 20$.
• To simplify $-3(2y-1)$, multiply each inner term by $-3$: $-3(2y) - 3(-1) = -6y + 3$.
• To simplify $(a-5)6$, distribute the 6 from the right: $a(6) - 5(6) = 6a - 30$.

Explanation

The distributive law lets you multiply a number outside parentheses by each term inside. It's like sharing the outside number with every term in the group through multiplication. This is essential when you can't combine the terms inside first.

Property

Steps for Solving Linear Equations.

1. Use the distributive law to remove any parentheses.
2. Combine like terms on each side of the equation.
3. By adding or subtracting the same quantity on both sides of the equation, get all the variable terms on one side and all the constant terms on the other.
4. Divide both sides by the coefficient of the variable to obtain an equation of the form $x = a$.

Examples

• Solve $3(x-4) = 9$. First, distribute the 3 to get $3x - 12 = 9$. Add 12 to both sides to get $3x = 21$. Finally, divide by 3 to find $x = 7$.
• Solve $5(y+1) = 2y - 4$. Distribute to get $5y+5 = 2y-4$. Subtract $2y$ from both sides, then subtract 5 from both sides to get $3y = -9$. Divide by 3 to find $y=-3$.
• Solve $25 - 4x = 2x - 5(2-x)$. Distribute to get $25 - 4x = 2x - 10 + 5x$. Combine like terms to get $25 - 4x = 7x - 10$. Add $4x$ to both sides, then add 10 to both sides to get $35 = 11x$, so $x = \frac{35}{11}$.

Explanation

When an equation has parentheses, first use the distributive law to clear them. After that, tidy up by combining like terms on each side. This simplifies the equation, making it easier to isolate the variable and find your solution.

Property

To build and simplify an algebraic expression from a description, first translate the words into variables and operations. If the expression involves multiplying a quantity in parentheses, use the distributive law to expand it. Finally, combine any like terms to get the simplest form. For example, for a rectangle with perimeter $P = 2l + 2w$, if the length $l$ is $2w-3$, we substitute to get $P = 2(2w - 3) + 2w$.

Examples

• A rectangle's length is 5 feet more than twice its width $w$. The perimeter is $P = 2(2w+5) + 2w$. Simplifying gives $P = 4w + 10 + 2w = 6w + 10$.
• A large pizza costs 5 dollars more than a medium pizza, $m$. If you buy 3 large pizzas, the total cost is $3(m+5)$. Using the distributive law, this simplifies to $3m + 15$.
• Ben is twice as old as his sister, $s$. In 4 years, Ben's age will be $2s+4$, and his sister's age will be $s+4$. The sum of their ages in 4 years will be $(2s+4) + (s+4) = 3s + 8$.

Explanation

The distributive law is a key tool for turning word problems into math. It helps you set up a formula based on a description and then simplify it into a clean, usable expression for solving problems.