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# Distributive Property for Multi-Step Equations

## a(x + b) = c

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Solve Equations with the Distributive Property

Sandra and Janet are going to the movies and meeting up with their friend Karen. The girls spent $42.75 in total for 3 movie tickets and 3 bags of popcorn that cost$5.25 each. How much money did the three girls spend on the movie tickets?

In this concept, you will learn to solve equations with the distributive property.

### Distributive Property

The distributive property states that when a factor is multiplied by the sum of two numbers, you can multiply each of the two numbers by that factor and then add them.

Let’s see how the distributive property can help you solve some multi-step equations.

Solve for \begin{align*}k\end{align*} in the following equation.

\begin{align*}5(3+k)=45\end{align*}

First, apply the distributive property to the left side of the equation. Multiply each of the two numbers inside the parentheses by 5 and then add those products.

\begin{align*}\begin{array}{rcl} 5(3+k) & = & 45\\ (5 \times 3) + (5 \times k) & = & 45\\ 15 + 5k & = & 45 \end{array}\end{align*}

Next, solve as you would solve any two-step equation. To isolate \begin{align*}5k\end{align*} on the left side of the equation, subtract 15 from both sides.

\begin{align*}\begin{array}{rcl} 15 + 5k & = & 45\\ 15 -15 + 5k & = & 45-15\\ 5k & = & 30 \end{array}\end{align*}

Then, divide both sides by 5 to solve for \begin{align*}k\end{align*}.

\begin{align*}\begin{array}{rcl} 5k & = & 30\\ \frac{5k}{5} & = & \frac{30}{5}\\ k & = & 6 \end{array}\end{align*}

Let’s look at another example.

Solve for \begin{align*}y\end{align*} in the following equation.

\begin{align*}2(y-9)=40\end{align*}

First, apply the distributive property to the left side of the equation. Multiply each of the two numbers inside the parentheses by 2 and then combine those products.

\begin{align*}\begin{array}{rcl} 2(y-9) & = & 45\\ (2 \times y) - (2 \times 9) & = & 45\\ 2y - 18 & = & 40 \end{array}\end{align*}

Next, solve as you would solve any two-step equation. To isolate \begin{align*}2y\end{align*} on the left side of the equation, add 18 from both sides.

\begin{align*}\begin{array}{rcl} 2y - 18 & = & 40\\ 2y -18 + 18 & = & 40-18\\ 2y & = & 58 \end{array}\end{align*}

Then, divide both sides by 2 to solve for \begin{align*}y\end{align*}.

\begin{align*}\begin{array}{rcl} 2y & = & 58\\ \frac{2y}{2} & = & \frac{58}{2}\\ y & = & 29 \end{array}\end{align*}

### Examples

#### Example 1

Remember the trip to the movies?

Sandra, Janet and Karen went to the movies and spent $42.75 and bought 3 tickets and 3 popcorns that cost$5.25 each. You want to find out the price of the movie tickets.

First, write your equation. Each girl bought a movie ticket and a bag of popcorn for $5.25. You also know that the total bill was$42.75. Let \begin{align*}x\end{align*} represent the price of each movie ticket.

\begin{align*}3(x+5.25)=42.75\end{align*}

Next, apply the distributive property to the left side of the equation. Multiply each of the two numbers inside the parentheses by 3 and then add those products.

\begin{align*}\begin{array}{rcl} 3(x+5.25) & = & 42.75\\ (3 \times x) + (3 \times 5.25) & = & 42.75\\ 3x + 15.75 & = & 42.75 \end{array}\end{align*}

Next, solve as you would solve any two-step equation. To isolate \begin{align*}3x\end{align*} on the left side of the equation, subtract 15.75 from both sides.

\begin{align*}\begin{array}{rcl} 3x +15.75 & = & 42.75\\ 3x + 15.75 - 15.75 & = & 42.75-15.75\\ 3x & = & 27 \end{array}\end{align*}

Then, divide both sides by 3 to solve for \begin{align*}x\end{align*}.

\begin{align*}\begin{array}{rcl} 3x & = & 27\\ \frac{3x}{3} & = & \frac{27}{3}\\ x & = & 9 \end{array}\end{align*}

Each movie ticket costs \$9.00.

#### Example 2

Solve for \begin{align*}x\end{align*} in the following equation.

\begin{align*}3(3-x)=12\end{align*}

First, apply the distributive property to the left side of the equation. Multiply each of the two numbers inside the parentheses by 2 and then combine those products.

\begin{align*}\begin{array}{rcl} 3(3-x) & = & 12\\ (3 \times 3) - (3 \times x) & = & 12\\ 9 - 3x & = & 12 \end{array}\end{align*}

Next, solve as you would solve any two-step equation. To isolate \begin{align*}3x\end{align*} on the left side of the equation, subtract 9 from both sides.

\begin{align*}\begin{array}{rcl} 9 - 3x & = & 12\\ (9-9) -3x & = & 12-9\\ -3x & = & 3 \end{array}\end{align*}

Then, divide both sides by -3 to solve for \begin{align*}x\end{align*}.

\begin{align*}\begin{array}{rcl} -3x & = & 3\\ \frac{-3x}{-3} & = & \frac{3}{-3}\\ x & = & -1 \end{array}\end{align*}

#### Example 3

Solve for \begin{align*}x\end{align*} in the following equation.

\begin{align*}6(x+4)=42\end{align*}

First, apply the distributive property to the left side of the equation. Multiply each of the two numbers inside the parentheses by 6 and then add those products.

\begin{align*}\begin{array}{rcl} 6(x+4) & = & 42\\ (6 \times x) + (6 \times 4) & = & 42\\ 6x + 24 & = & 42 \end{array}\end{align*}

Next, solve as you would solve any two-step equation. To isolate \begin{align*}6x\end{align*} on the left side of the equation, subtract 24 from both sides.

\begin{align*}\begin{array}{rcl} 6x + 24 & = & 42\\ 6x + 24 - 24 & = & 42-24\\ 6x & = & 18 \end{array}\end{align*}

Then, divide both sides by 6 to solve for \begin{align*}x\end{align*}.

\begin{align*}\begin{array}{rcl} 6x & = & 18\\ \frac{6x}{6} & = & \frac{18}{6}\\ x & = & 3 \end{array}\end{align*}

#### Example 4

Solve for \begin{align*}y\end{align*} in the following equation.

\begin{align*}4(y-8)=16\end{align*}

First, apply the distributive property to the left side of the equation. Multiply each of the two numbers inside the parentheses by 4 and then combine those products.

\begin{align*}\begin{array}{rcl} 4(y-8) & = & 16\\ (4 \times y) - (4 \times 8) & = & 16\\ 4y - 32 & = & 16 \end{array}\end{align*}

Next, solve as you would solve any two-step equation. To isolate \begin{align*}4y\end{align*} on the left side of the equation, add 32 from both sides.

\begin{align*}\begin{array}{rcl} 4y - 32 & = & 16\\ 4y - 32 + 32 & = & 16+32\\ 4y & = & 48 \end{array}\end{align*}

Then, divide both sides by 4 to solve for \begin{align*}y\end{align*}.

\begin{align*}\begin{array}{rcl} 4y & = & 48\\ \frac{4y}{4} & = & \frac{48}{4}\\ y & = & 12 \end{array}\end{align*}

#### Example 5

Solve for \begin{align*}y\end{align*} in the following equation.

\begin{align*}12(x-2)=48\end{align*}

First, apply the distributive property to the left side of the equation. Multiply each of the two numbers inside the parentheses by 12 and then combine those products.

\begin{align*}\begin{array}{rcl} 12(x-2) & = & 48\\ (12 \times x) - (12 \times 2) & = & 48\\ 12x - 24 & = & 48 \end{array}\end{align*}

Next, solve as you would solve any two-step equation. To isolate \begin{align*}12x\end{align*} on the left side of the equation, add 24 from both sides.

\begin{align*}\begin{array}{rcl} 12x - 24 & = & 48\\ 12x - 24 + 24 & = & 48+24\\ 12x & = & 72 \end{array}\end{align*}

Then, divide both sides by 12 to solve for \begin{align*}x\end{align*}.

\begin{align*}\begin{array}{rcl} 12x & = & 72\\ \frac{12x}{12} & = & \frac{72}{12}\\ x & = & 6 \end{array}\end{align*}

### Review

Use the distributive property to solve each equation.

1. \begin{align*}2(x+3)=10\end{align*}
2. \begin{align*}5(x+4)=25\end{align*}
3. \begin{align*}9(x- 3) =27\end{align*}
4. \begin{align*}7(x+5)=70\end{align*}
5. \begin{align*}5(x- 6) =45\end{align*}
6. \begin{align*}8(y- 4) =40\end{align*}
7. \begin{align*}7(x+3)=-7\end{align*}
8. \begin{align*}8(x- 2) =8\end{align*}
9. \begin{align*}9(y+1)=90\end{align*}
10. \begin{align*}-3(y+4)=24\end{align*}
11. \begin{align*}-2 (y- 4) =16\end{align*}
12. \begin{align*}-4 (x- 1) =8\end{align*}
13. \begin{align*}9(y-4)=36\end{align*}
14. \begin{align*}7(y-3)=21\end{align*}
15. \begin{align*}-9 (y- 2) =27\end{align*}

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### Vocabulary Language: English

distributive property

The distributive property states that the product of an expression and a sum is equal to the sum of the products of the expression and each term in the sum. For example, $a(b + c) = ab + ac$.

factor

Factors are the numbers being multiplied to equal a product. To factor means to rewrite a mathematical expression as a product of factors.

Variable

A variable is a symbol used to represent an unknown or changing quantity. The most common variables are a, b, x, y, m, and n.