3.6: Solve Equations with the Distributive Property
Have you ever needed the distributive property to solve a problem? Well, Trevor has a dilemma. He is having difficulty figuring out this problem.
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
You already know that some number properties can help you solve equations.
The distributive property may also help you solve some equations.
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.
Multiplication can also be distributed over subtraction.
Here are two situations that show the distributive property.
Let's see how the distributive property can help us solve some multistep equations.
Solve for
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.
Now, solve as you would solve any twostep equation. To get
To get
The value of
Let's look at another one.
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.
Next, we solve this for
The value of
Example A
Solution:
Example B
Solution:
Example C
Solution:
Now let's go back to the dilemma at the beginning of the Concept.
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.
Next, we solve this twostep equation. Subtract 14 from both sides of the equation.
Now we can solve the onestep equation by dividing both sides of the equation by 7.
This is our final answer.
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
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
Now, solve as you would solve any twostep equation. We need to first get the term that includes a variable,
To get
The value of
Video Review
Khan Academy The Distributive Property
Practice
Directions: Use the distributive property to solve each equation.

2(x+3)=10 
5(x+4)=25 
9(x−3)=27 
7(x+5)=70 
5(x−6)=45 
8(y−4)=40  \begin{align*}7(x+3)=7\end{align*}
 \begin{align*}8(x2)=8\end{align*}
 \begin{align*}9(y+1)=90\end{align*}
 \begin{align*}3(y+4)=24\end{align*}
 \begin{align*}2(y4)=16\end{align*}
 \begin{align*}4(x1)=8\end{align*}
 \begin{align*}9(y4)=36\end{align*}
 \begin{align*}7(y3)=21\end{align*}
 \begin{align*}9(y2)= 27\end{align*}
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, .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
Here you'll learn to solve equations with the distributive property.
Concept Nodes:
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, .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.