# USING THE DISTRIBUTIVE PROPERTY

We can use the Distributive Property to remove the parentheses from an algebraic expression like 3(x + 5). Sometimes this is called “simplifying” or “expanding” the expression. Multiply the quantity in front of parentheses by each term within parentheses :

3(x + 5)  =  3 · x + 3 · 5

3(x + 5)  =  3x + 15.

## Examples

Example 1 :

Simplify the following expression :

5 - 3(7x + 8)

Solution :

=  5 - 3(7x + 8)

Use the distributive property.

=  5 - 3(7x) - 3(8)

Simplify.

=  5 - 21x - 24

=  -21x - 19

Example 2 :

Simplify the following expression :

-9a - (1/3)(- 3/4 - 2a/3 + 12)

Solution :

-9a - (1/3)(- 3/4 - 2a/3 + 12)

Use the distributive property.

=  -9a + 1/4 + 2a/9 - 4

Simplify.

=  -9a + 2a/9 + 1/4 - 4

=  -81a/9 + 2a/9 + 1/4 - 16/4

=  (-81a + 2a)/9 + (1 - 16)/4

=  -79a/9 - 15/4

Example 3 :

Simplify the following expression :

0.2(3b - 15c) + 6c

Solution :

0.2(3b - 15c) + 6c

Use the distributive property.

=  0.6b - 3c + 6c

Simplify.

=  0.6b + 3c

Example 4 :

Simplify the following expression :

(2/3)(6e + 9f -21g) - 7f

Solution :

=  (2/3)(6e + 9f - 21g) - 7f

Use the distributive property.

=  4e + 6f - 14g - 7f

Simplify.

=  4e - f - 14g

Example 5 :

James is selling tickets for a high school band concert. The band gets to keep 25% of the money he collects from ticket sales to put toward new uniforms. Write an expression to represent how much the band gets to keep. Solution :

Let a represent the number of adult tickets he sells.

Let y represent the number of youth tickets he sells.

The expression 16.60a + 12.20y represents the amount of money James collects from ticket sales.

Write 25% as a decimal :

0.25

Write an expression to represent 25% of the money he collects :

0.25(16.60a + 12.20y)

That is, 25% adult ticket sales and youth ticket sales.

Use the Distributive Property to simplify the expression.

=  0.25(16.60a) + 0.25(12.20y)

=  4.15a + 3.05y

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