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Simplifying Expressions Using the Distributive Property

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Simplifying Expressions Using the Distributive Property

Suppose three friends are going to the movies. They each need \(\text{\$9.25};\) that is, \(9\) dollars and \(1\) quarter. How much money do they need all together? You can think about the dollars separately from the quarters.

Simplifying Expressions Using the Distributive Property

Simplifying Expressions Using the Distributive Property

They need \(3\) times \(\text{\$9},\) so \(\text{\$27},\) and \(3\) times \(1\) quarter, so \(75\) cents. In total, they need \(\text{\$27.75}.\)

If you think about doing the math in this way, you are using the Distributive Property.

Definition: Distributive Property

If \(a,b,c\) are real numbers, then

\(a\left(b+c\right)=ab+ac\)

Back to our friends at the movies, we could show the math steps we take to find the total amount of money they need like this:

\[\begin{array}{ccc} & 3(9.25) & \\ 3(9 & + & 0.25) \\ 3(9) & + & 3(0.25) \\ 27 &  + & 0.75 \\ & 27.75 & \end{array}\]

In algebra, we use the Distributive Property to remove parentheses as we simplify expressions. For example, if we are asked to simplify the expression \(3\left(x+4\right),\) the order of operations says to work in the parentheses first. But we cannot add \(x\) and \(4,\) since they are not like terms. So we use the Distributive Property, as shown below.

Optional Video: Model Distribution

Example

Simplify: \(3\left(x+4\right).\)

Solution

 \(3\left(x+4\right)\)
Distribute.\(3·x+3·4\)
Multiply.\(3x+12\)

Some students find it helpful to draw in arrows to remind them how to use the Distributive Property. Then the first step in the example above would look like this:

Simplifying Expressions Using the Distributive PropertySimplifying Expressions Using the Distributive Property

Example

Simplify: \(6\left(5y+1\right).\)

Solution

 Simplifying Expressions Using the Distributive Property
Distribute.Simplifying Expressions Using the Distributive Property
Multiply.Simplifying Expressions Using the Distributive Property

The distributive property can be used to simplify expressions that look slightly different from \(a\left(b+c\right).\) Here are two other forms.

Using the Distributive Property

If \(a,b,c\) are real numbers, then

\(a\left(b+c\right)=ab+ac\)

Other forms

\(a\left(b-c\right)=ab-ac\)

\(\left(b+c\right)a=ba+ca\)

Example

Simplify: \(2\left(x-3\right).\)

Solution

 Simplifying Expressions Using the Distributive Property
Distribute.Simplifying Expressions Using the Distributive Property
Multiply.Simplifying Expressions Using the Distributive Property

Do you remember how to multiply a fraction by a whole number? We’ll need to do that in the next two examples.

Example

Simplify: \(\frac{3}{4}\left(n+12\right).\)

Solution

 Simplifying Expressions Using the Distributive Property
Distribute.Simplifying Expressions Using the Distributive Property
Simplify.Simplifying Expressions Using the Distributive Property

Example

Simplify: \(8\left(\frac{3}{8}x+\frac{1}{4}\right).\)

Solution

 Simplifying Expressions Using the Distributive Property
Distribute.Simplifying Expressions Using the Distributive Property
Multiply.Simplifying Expressions Using the Distributive Property

Using the Distributive Property as shown in the next example will be very useful when we solve money applications later.

Example

Simplify: \(100\left(0.3+0.25q\right).\)

Solution

 Simplifying Expressions Using the Distributive Property
Distribute.Simplifying Expressions Using the Distributive Property
Multiply.Simplifying Expressions Using the Distributive Property

In the next example we’ll multiply by a variable. We’ll need to do this in a later tutorial.

Example

Simplify: \(m\left(n-4\right).\)

Solution

 Simplifying Expressions Using the Distributive Property
Distribute.Simplifying Expressions Using the Distributive Property
Multiply.Simplifying Expressions Using the Distributive Property

Notice that we wrote \(m·4\phantom{\rule{0.2em}{0ex}}\text{as}\phantom{\rule{0.2em}{0ex}}4m.\) We can do this because of the Commutative Property of Multiplication. When a term is the product of a number and a variable, we write the number first.

The next example will use the ‘backwards’ form of the Distributive Property, \(\left(b+c\right)a=ba+ca.\)

Example

Simplify: \(\left(x+8\right)p.\)

Solution

 Simplifying Expressions Using the Distributive Property
Distribute.Simplifying Expressions Using the Distributive Property

When you distribute a negative number, you need to be extra careful to get the signs correct.

Example

Simplify: \(-2\left(4y+1\right).\)

Solution

 Simplifying Expressions Using the Distributive Property
Distribute.Simplifying Expressions Using the Distributive Property
Simplify.Simplifying Expressions Using the Distributive Property

Example

Simplify: \(-11\left(4-3a\right).\)

Solution

 Simplifying Expressions Using the Distributive Property
Distribute.Simplifying Expressions Using the Distributive Property
Multiply.Simplifying Expressions Using the Distributive Property
Simplify.Simplifying Expressions Using the Distributive Property

You could also write the result as \(33a-44.\) Do you know why?

In the next example, we will show how to use the Distributive Property to find the opposite of an expression. Remember, \(-a=-1·a.\)

Example

Simplify: \(-\left(y+5\right).\)

Solution

 Simplifying Expressions Using the Distributive Property
Multiplying by −1 results in the opposite.Simplifying Expressions Using the Distributive Property
Distribute.Simplifying Expressions Using the Distributive Property
Simplify.Simplifying Expressions Using the Distributive Property
Simplify.Simplifying Expressions Using the Distributive Property

Sometimes we need to use the Distributive Property as part of the order of operations. Start by looking at the parentheses. If the expression inside the parentheses cannot be simplified, the next step would be multiply using the distributive property, which removes the parentheses. The next two examples will illustrate this.

Example

Simplify: \(8-2\left(x+3\right).\)

Solution

 Simplifying Expressions Using the Distributive Property
Distribute.Simplifying Expressions Using the Distributive Property
Multiply.Simplifying Expressions Using the Distributive Property
Combine like terms.Simplifying Expressions Using the Distributive Property

Example

Simplify: \(4\left(x-8\right)-\left(x+3\right).\)

Solution

 Simplifying Expressions Using the Distributive Property
Distribute.Simplifying Expressions Using the Distributive Property
Combine like terms.Simplifying Expressions Using the Distributive Property

Optional Video: The Distributive Property

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