1.6.3: Expressions and the Distributive Property

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Expressions and the Distributive Property

Karen's biology class is going on a trip to the beach. Karen is in charge of keeping track of who has paid for the trip and who still needs to pay. There are 50 students going on the trip. Each student must pay $5 for transportation, $4.50 for lunch, and $3 for a t-shirt. How could Karen write and simplify a numerical expression using the distributive property to determine how much money should be collected total?

In this concept, you will learn how to use the distributive property to write and evaluate equivalent numerical and variable expressions.

Using the Distributive Property with Expressions

The distributive property states that if 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 to produce the same result. In symbols, the distributive property says that for numbers a, b and c:

a(b+c)=ab+ac

Anytime you see a number outside of a set of parentheses within an expression, you can use the distributive property to help you to simplify by writing an equivalent expression without parentheses.

Let's look at an example.

5(2+3)

You want to simplify by writing an equivalent expression without parentheses.

First, multiply the 5 by each of the terms inside the parentheses.

5(2+3)=5(2)+5(3)

Next, simplify each part of the expression and then add.

5(2)+5(3)=10+15=25

The answer is 5(2+3)=25.

The distributive property also works if there is subtraction inside the parentheses. This is because subtracting a positive is just like adding a negative.

Here is an example.

3(3−2)

You want to simplify by writing an equivalent expression without parentheses.

First, multiply the 3 by each of the terms inside the parentheses.

3(3−2)=3(3)−3(2)

Next, simplify each part of the expression and then subtract.

3(3)−3(2)=9−6=3

The answer is 3(3−2)=3.

Notice this is the same answer that you would have gotten had you first subtracted the numbers inside the parentheses and then multiplied:

3(3−2)=3(1)=3

The distributive property isn't as necessary in situations where all parts of your expressions are numbers like in the previous two examples. However, it becomes very useful with variable expressions that contain variables. The distributive property helps you to get rid of the parentheses in variable expressions.

Here is an example.

4(x+3)

First, write an equivalent expression without parentheses. Multiply the 4 by each term inside the parentheses.

4(x+3)=4(x)+4(3)

Next, simplify each part of the expression.

4(x)+4(3)=4x+12

The answer is 4(x+3)=4x+12.

Examples

Example $\PageIndex{1}$

Earlier, you were given a problem about Karen and her biology class trip to the beach.

Each of the 50 students going on the trip must pay $5 for transportation, $4.50 for lunch, and $3 for a t-shirt. Karen wants to know how much money will be collected in total.

Each of the 50 students going on the trip has to pay 5+4.50+3 for transportation, lunch, and a t-shirt. That means the total amount of money collected will be

50(5+4.50+3)

One way to complete this calculation is by using the distributive property.

Solution

First, write an equivalent expression without parentheses. Multiply the 5 by each of the terms inside the parentheses.

50(5+4.50+3)=50(5)+50(4.50)+50(3)

Next, simplify each part of the expression and then add.

50(5)+50(4.50)+50(3)=250+225+150=625

The answer is 625.

In total, $625 will be collected for the trip to the beach.

Example $\PageIndex{1}$

Liam has a rectangular backyard that is 20 yards long and 18 yards wide. He wants to use a part of his yard that is 20 yards by 8 yards for a vegetable garden. If he does this, what will be the area of the section of the yard that will not be used as a garden?

Solution

First, remember that to find the area of a rectangle you need to multiply length times width.

Next, figure out the length and width of the part of the backyard not used as a garden.

The length of the part of the backyard not used as a garden is 20 yards.
The width of the part of the backyard not used as a garden is (18 yards - 8 yards).

Now, write out the product of length times width.

Area=20(18−8)

One way to complete this multiplication is by using the distributive property. Multiply 20 by each term inside the parentheses.

Area=20(18)−20(8)

Now simplify each part of the expression and then subtract.

Area=360−160=200

The answer is that the area of the backyard not used as a garden is 200 square yards.

Example $\PageIndex{1}$

Simplify using the distributive property.

6(5+2)

Solution

First, write an equivalent expression without parentheses. Multiply the 6 by each of the terms inside the parentheses.

6(5+2)=6(5)+6(2)

Next, simplify each part of the expression and then add.

6(5)+6(2)=30+12=42

The answer is 6(5+2)=42.

Example $\PageIndex{1}$

Simplify using the distributive property.

3(x−5)

Solution

First, write an equivalent expression without parentheses. Multiply the 3 by each term inside the parentheses.

3(x−5)=3(x)−3(5)

Next, simplify each part of the expression.

3(x)−3(5)=3x−15

The answer is 3(x−5)=3x−15.

Example $\PageIndex{1}$

Simplify using the distributive property.

8(9+y)

Solution

First, write an equivalent expression without parentheses. Multiply the 8 by each term inside the parentheses.

8(9+y)=8(9)+8(y)

Next, simplify each part of the expression.

8(9)+8(y)=72+8y

The answer is 8(9+y)=72+8y.

Review

Use the distributive property to write an equivalent expression for each numerical expression.

6(3+4)
5(4+1)
12(3+5)
6(7+8)
2(4+5)
3(5−2)
6(7−3)
5(4−2)
7(5−1)
6(9−3)

Use the distributive property to write an equivalent expression for each variable expression.

5(x+3)
6(y−2)
5(x+9)
8(a+b)
7(x−y)

Review (Answers)

To see the Review answers, open this PDF file and look for section 4.16.

Vocabulary

Term	Definition
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.
Equivalent	Equivalent means equal in value or meaning.

Additional Resources

Video:

Activity: Expressions and the Distributive Property Discussion Questions

Practice: Expressions and the Distributive Property

Real World Application: Party Planning