Let's say that a piece of gift-wrapping paper had a length of 5 feet and a width of \begin{align*}x\end{align*}

### Expressions and the Distributive Property

The Distributive Property is a way of distributing a value equally between two or more other values contained in parentheses. The **Di****stributive Property **states that for any real numbers \begin{align*}M, \ N,\end{align*}

\begin{align*}& M(N+K)= MN+MK\\
& M(N-K)= MN-MK\end{align*}

Suppose that at the end of the school year, an elementary school teacher makes a little gift bag for each of his students. Each bag contains one class photograph, two party favors, and five pieces of candy. The teacher will distribute the bags among his 28 students. How many of each item does the teacher need?

You could begin to solve this problem by deciding your variables.

Let \begin{align*}p=photograph, \ f=favors,\end{align*}

Next you can write an expression to represent the situation: \begin{align*}p + 2f + 5c.\end{align*}

There are 28 students in class, so the teacher needs to repeat the bag 28 times. An easier way to write this is \begin{align*}28 \cdot (p + 2f + 5c).\end{align*}

We can omit the multiplication symbol and write \begin{align*}28(p + 2f + 5c)\end{align*}

Therefore, the teacher needs \begin{align*}28p + 28(2f) + 28(5c)\end{align*}

The teacher needs 28 photographs, 56 favors, and 140 pieces of candy to complete the end-of-year gift bags.

When you multiply an algebraic expression by another expression, you apply the distributive property.

#### Let's simplify the following expressions:

- \begin{align*}11(2 + 6)\end{align*}

Using the Order of Operations: \begin{align*}11(2 + 6) = 11(8)= 88.\end{align*}

Using the Distributive Property: \begin{align*}11(2 + 6) = 11(2) + 11(6)= 22 + 66 = 88.\end{align*}

Notice that regardless of the method, the answer is the same.

- \begin{align*}7(3x - 5).\end{align*}

Think of this expression as seven groups of \begin{align*}(3x -5)\end{align*}. You could write this expression seven times and add all the like terms.

\begin{align*}(3x-5)+(3x-5)+(3x-5)+(3x-5)+(3x-5)+(3x-5)+(3x-5)=21x-35\end{align*}

Apply the Distributive Property.

\begin{align*}7(3x-5)= 7(3x)+7(-5)= 21x-35\end{align*}

### Examples

#### Example 1

Earlier, you were told that a piece of gift-wrapping paper had a length of 5 feet and a width of \begin{align*}x\end{align*} feet. When wrapping a present, you cut off a vertical strip of 1 foot so that the width of the piece of gift-wrapping paper is decreased by 1 foot. What would the area of the remaining paper be in square feet? Could you write this expression in simplified form?

The height of the remaining piece of wrapping paper is still 5 since you did not cut off a strip horizontally. If the original width of the wrapping paper is \begin{align*}x\end{align*} feet, then the width of the remaining piece of wrapping paper is now \begin{align*}x-1\end{align*} since the width was decreased by 1 foot. To find the area, multiply the width and the height:

\begin{align*}5(x-1)\end{align*} To simplify this expression, apply the Distributive Property:

\begin{align*}5(x-1)=5x-5\end{align*} The area of the remaining paper is \begin{align*}5x-5\end{align*} square feet.

#### Example 2

Simplify \begin{align*}\frac{2}{7} (3y^2 - 11).\end{align*}

Apply the Distributive Property.

\begin{align*}&\frac{2}{7} (3y^2 + -11)= \frac{2}{7} (3y^2) + \frac{2}{7}(-11)=\\ &\frac{2}{7} \frac{(3y^2)}{1} + \frac{2}{7}\frac{(-11)}{1}=\frac{2 \times 3y^2}{7 \times 1} + \frac{2\times (-11)}{7\times 1}=\\ &\frac{6y^2}{7}+\frac{-22}{7}=\frac{6y^2}{7}-\frac{22}{7}\end{align*}

### Review

Use the Distributive Property to simplify the following expressions.

- \begin{align*}(x + 4) - 2(x + 5)\end{align*}
- \begin{align*}\frac{1}{2}(4z + 6)\end{align*}
- \begin{align*}(4 + 5)-(5 + 2)\end{align*}
- \begin{align*}(x + 2 + 7)\end{align*}
- \begin{align*}0.25 (6q + 32)\end{align*}
- \begin{align*}y(x + 7)\end{align*}
- \begin{align*}-4.2(h - 11)\end{align*}
- \begin{align*}13x(3y + z)\end{align*}
- \begin{align*}\frac{1}{2}(x - y) - 4\end{align*}
- \begin{align*}0.6(0.2x + 0.7)\end{align*}
- \begin{align*}(2 - j)(-6)\end{align*}
- \begin{align*}4(m + 7) -6(4 - m)\end{align*}
- \begin{align*}-5(y - 11) + 2y\end{align*}

**Mixed Review**

- Translate the following into an inequality: Jacob wants to go to Chicago for his class trip. He needs at least $244 for the bus, hotel stay, and spending money. He already has $104. How much more does he need to pay for his trip?
- Underline the math verb(s) in this sentence: 6 times a number is 4 less than 16.
- Draw a picture to represent \begin{align*}3 \frac{3}{4}\end{align*}.
- Determine the change in \begin{align*}y\end{align*} in the equation \begin{align*}y = \frac{1}{6} x-4\end{align*} between \begin{align*}x=3\end{align*} and \begin{align*}x=9\end{align*}.

### Review (Answers)

To see the Review answers, open this PDF file and look for section 2.8.