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Distributive Property for Multi-Step Equations

a(x + b) = c

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Solve Equations with the Distributive Property
License: CC BY-NC 3.0

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*}k in the following equation.

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

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*}5(3+k)(5×3)+(5×k)15+5k===454545

Next, solve as you would solve any two-step equation. To isolate \begin{align*}5k\end{align*}5k 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*}15+5k1515+5k5k===45451530

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

\begin{align*}\begin{array}{rcl} 5k & = & 30\\ \frac{5k}{5} & = & \frac{30}{5}\\ k & = & 6 \end{array}\end{align*}5k5k5k===303056

The answer is 6.

Let’s look at another example.

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

\begin{align*}2(y-9)=40\end{align*}2(y9)=40

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*}2(y9)(2×y)(2×9)2y18===454540 

Next, solve as you would solve any two-step equation. To isolate \begin{align*}2y\end{align*}2y 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*}2y182y18+182y===40401858

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

\begin{align*}\begin{array}{rcl} 2y & = & 58\\ \frac{2y}{2} & = & \frac{58}{2}\\ y & = & 29 \end{array}\end{align*}2y2y2y===5858229

The answer is 29.

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*}x represent the price of each movie ticket.

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

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*}3(x+5.25)(3×x)+(3×5.25)3x+15.75===42.7542.7542.75

Next, solve as you would solve any two-step equation. To isolate \begin{align*}3x\end{align*}3x 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*}3x+15.753x+15.7515.753x===42.7542.7515.7527

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

\begin{align*}\begin{array}{rcl} 3x & = & 27\\ \frac{3x}{3} & = & \frac{27}{3}\\ x & = & 9 \end{array}\end{align*}3x3x3x===272739

The answer is 9.

Each movie ticket costs $9.00.

Example 2

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

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

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*}3(3x)(3×3)(3×x)93x===121212 

Next, solve as you would solve any two-step equation. To isolate \begin{align*}3x\end{align*}3x 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*}93x(99)3x3x===121293

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

\begin{align*}\begin{array}{rcl} -3x & = & 3\\ \frac{-3x}{-3} & = & \frac{3}{-3}\\ x & = & -1 \end{array}\end{align*}3x3x3x===3331

The answer is -1.

Example 3

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

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

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*}6(x+4)(6×x)+(6×4)6x+24===424242 

Next, solve as you would solve any two-step equation. To isolate \begin{align*}6x\end{align*}6x 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*}6x+246x+24246x===42422418

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

\begin{align*}\begin{array}{rcl} 6x & = & 18\\ \frac{6x}{6} & = & \frac{18}{6}\\ x & = & 3 \end{array}\end{align*}6x6x6x===181863

The answer is 3.

Example 4

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

\begin{align*}4(y-8)=16\end{align*}4(y8)=16

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*}4(y8)(4×y)(4×8)4y32===161616

Next, solve as you would solve any two-step equation. To isolate \begin{align*}4y\end{align*}4y 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*}4y324y32+324y===1616+3248

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

\begin{align*}\begin{array}{rcl} 4y & = & 48\\ \frac{4y}{4} & = & \frac{48}{4}\\ y & = & 12 \end{array}\end{align*}4y4y4y===4848412

The answer is 12.

Example 5

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

\begin{align*}12(x-2)=48\end{align*}12(x2)=48

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*}12(x2)(12×x)(12×2)12x24===484848 

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*}

The answer is 6.

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*}

Review (Answers)

To see the Review answers, open this PDF file and look for section 3.6. 

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Vocabulary

TermDefinition
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.

Image Attributions

  1. [1]^ License: CC BY-NC 3.0

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