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3.6: Solve Equations with the Distributive Property

Difficulty Level: At Grade Created by: CK-12
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Practice Distributive Property for Multi-Step Equations
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Estimated22 minsto complete
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Have you ever needed the distributive property to solve a problem? Well, Trevor has a dilemma. He is having difficulty figuring out this problem.

7(x+2)=28\begin{align*}7(x+2)=28\end{align*}

Do you know how to solve this equation? To figure it out, you will have to apply the distributive property. Take a look at this Concept and you will know how to solve this equation by the end of it.

Guidance

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

7×(4+k)=(7×4)+(7×k)=28+7k\begin{align*}7 \times (4+k) = (7 \times 4) + (7 \times k) = 28 + 7k\end{align*}

2(a+3)=(2×a)+(2×3)=2a+6\begin{align*}2(a + 3) = (2 \times a) + (2 \times 3) = 2a + 6\end{align*}

Multiplication can also be distributed over subtraction.

Here are two situations that show the distributive property.

7×(4k)=(7×4)(7×k)=287k\begin{align*}7 \times (4 - k) = (7 \times 4) - (7 \times k) = 28 - 7k\end{align*}

2(a3)=(2×a)(2×3)=2a6\begin{align*}2(a - 3) = (2 \times a) - (2 \times 3) = 2a - 6\end{align*}

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

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

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.

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

Now, solve as you would solve any two-step equation. To get 5k\begin{align*}5k\end{align*} by itself on one side of the equation, subtract 15 from both sides.

15+5k1515+5k0+5k5k=45=4515=30=30\begin{align*}15+5k &= 45\\ 15-15+5k &= 45-15\\ 0+5k &= 30\\ 5k &= 30\end{align*}

To get k\begin{align*}k\end{align*} by itself on one side of the equation, divide both sides by 5.

5k5k51kk=30=305=6=6\begin{align*}5k &= 30\\ \frac{5k}{5} &= \frac{30}{5}\\ 1k &= 6\\ k &= 6\end{align*}

The value of k\begin{align*}k\end{align*} is 6.

Let's look at another one.

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

Now we can distribute the two by multiplying it by both of the terms inside the parentheses. Notice that the second term has a subtraction sign in front of it. Remember to include that sign when we multiply.

2y18=40\begin{align*}2y - 18 = 40\end{align*}

Next, we solve this for y\begin{align*}y\end{align*} as we would with any two step equation.

2y182y18+182yy=40=40+18=58=29\begin{align*}2y - 18 &= 40\\ 2y - 18 + 18 &= 40 + 18\\ 2y &= 58\\ y &= 29\end{align*}

The value of y\begin{align*}y\end{align*} is 29.

Example A

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

Solution:  x=3\begin{align*}x = 3\end{align*}

Example B

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

Solution:  y=12\begin{align*}y = 12\end{align*}

Example C

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

Solution:  x=6\begin{align*}x = 6\end{align*}

Now let's go back to the dilemma at the beginning of the Concept.

7(x+2)=28\begin{align*}7(x+2)=28\end{align*}

This is the equation that needs to be solved.

First, we have to simplify the left side of the equation by getting rid of the parentheses. We do this by multiply both of the terms inside the parentheses by 7.

7x+14=28\begin{align*}7x + 14 = 28\end{align*}

Next, we solve this two-step equation. Subtract 14 from both sides of the equation.

7x+1414=2814\begin{align*}7x + 14 - 14 = 28 - 14\end{align*}

7x=14\begin{align*}7x = 14\end{align*}

Now we can solve the one-step equation by dividing both sides of the equation by 7.

x=2\begin{align*}x = 2\end{align*}

Vocabulary

Distributive Property
states that you can multiply a term outside of a set of parentheses with the terms inside the parentheses to simplify the set of parentheses.

Guided Practice

Here is one for you to try on your own.

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

Solution

Apply the distributive property to the left side of the equation. Multiply each of the two numbers inside the parentheses by 3 and then subtract those products. It may help you to remember that x=1x\begin{align*}x = 1x\end{align*}.

3(3x)3(31x)(33)(31x)93x=12=12=12=12\begin{align*}3(3-x) &= 12\\ 3(3 - 1x) &= 12\\ (3 \cdot 3) - (3 \cdot 1x) &= 12\\ 9 - 3x &= 12\end{align*}

Now, solve as you would solve any two-step equation. We need to first get the term that includes a variable, 3x\begin{align*}3x\end{align*}, by itself on one side of the equation. In the equation, 3x\begin{align*}3x\end{align*} is subtracted from 9. Subtracting 93x\begin{align*}9 - 3x\end{align*} is the same as adding 9+(3x)\begin{align*}9 + (-3x)\end{align*}. Rewrite the left side of the equation to show that 9 is being added to 3x\begin{align*}-3x\end{align*}, and then subtract 9 from both sides.

93x9+(3x)99+(3x)0+(3x)3x=12=12=129=3=3\begin{align*}9 - 3x &= 12\\ 9 + (-3x) &= 12\\ 9 - 9 + (-3x) &= 12 - 9\\ 0 + (-3x) &= 3\\ -3x &= 3\end{align*}

To get x\begin{align*}x\end{align*} by itself on one side of the equation, divide both sides by -3. You will need to use what you know about dividing integers to help you. For example, you know that when you divide two negative integers, the quotient will be positive. Since you know that 3÷(3)=1\begin{align*}-3 \div (-3) = 1\end{align*}, you also know that 3x÷(3)=1x\begin{align*}-3x \div (-3) = 1x\end{align*}. To review how to compute with integers, look back at Lessons 2.5 and 2.6.

3x3x31xx=3=33=1=1\begin{align*}-3x &= 3\\ \frac{-3x}{-3} &= \frac{3}{-3}\\ 1x &= -1\\ x &= -1\end{align*}

The value of x\begin{align*}x\end{align*} is -1.

Practice

Directions: Use the distributive property to solve each equation.

1. 2(x+3)=10\begin{align*}2(x+3)=10\end{align*}
2. 5(x+4)=25\begin{align*}5(x+4)=25\end{align*}
3. 9(x3)=27\begin{align*}9(x-3)=27\end{align*}
4. 7(x+5)=70\begin{align*}7(x+5)=70\end{align*}
5. 5(x6)=45\begin{align*}5(x-6)=45\end{align*}
6. 8(y4)=40\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*}

Vocabulary Language: English

distributive property

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

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

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.

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