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# 5.3: Applications Using Linear Models

Difficulty Level: At Grade Created by: CK-12
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Practice Applications Using Linear Models

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### Applications using Linear Models

Let’s solve some word problems where we need to write the equation of a straight line in point-slope form.

#### Real-World Application: Moving Trucks

Marciel rented a moving truck for the day. Marciel only remembers that the rental truck company charges $40 per day and some number of cents per mile. Marciel drives 46 miles and the final amount of the bill (before tax) is$63. What is the amount per mile the truck rental company charges? Write an equation in point-slope form that describes this situation. How much would it cost to rent this truck if Marciel drove 220 miles?

Let’s define our variables:

xy=distance in miles=cost of the rental truck\begin{align*}x &= \text{distance in miles}\!\\ y &= \text{cost of the rental truck}\end{align*}

Peter pays a flat fee of 40 for the day; this is the y\begin{align*}y-\end{align*}intercept. He pays63 for 46 miles; this is the coordinate point (46,63).

Start with the point-slope form of the line: yy0=m(xx0)\begin{align*}y-y_0=m(x-x_0)\end{align*}

Plug in the coordinate point: 63y0=m(46x0)\begin{align*}63-y_0=m(46-x_0)\end{align*}

Plug in the point (0, 40): 6340=m(460)\begin{align*}63-40=m(46-0)\end{align*}

Solve for the slope: 23=46mm=2346=0.5\begin{align*}23=46m \rightarrow m=\frac{23}{46}=0.5\end{align*}

The slope is 0.5 dollars per mile, so the truck company charges 50 cents per mile (0.5 = 50 cents). Plugging in the slope and the y\begin{align*}y-\end{align*}intercept, the equation of the line is y=0.5x+40\begin{align*}y=0.5x+40\end{align*}. To find out the cost of driving the truck 220 miles, we substitute 220 for x\begin{align*}x\end{align*} to get y40=0.5(220)y=150\begin{align*}y-40=0.5(220) \Rightarrow y=\150\end{align*}.

Driving 220 miles would cost $150. #### Real-World Application: Sales Commission Anne got a job selling window shades. She receives a monthly base salary and a$6 commission for each window shade she sells. At the end of the month she adds up sales and she figures out that she sold 200 window shades and made 2500. Write an equation in point-slope form that describes this situation. How much is Anne’s monthly base salary? Let’s define our variables: xy=number of window shades sold=Anne's earnings\begin{align*}x &= \text{number of window shades sold}\!\\ y &= \text{Anne's earnings}\end{align*} We see that we are given the slope and a point on the line: Nadia gets6 for each shade, so the slope is 6.

She made 2500 when she sold 200 shades, so the point is (200, 2500). Start with the point-slope form of the line: yy0=m(xx0)\begin{align*}y-y_0=m(x-x_0)\end{align*} Plug in the slope: yy0=6(xx0)\begin{align*}y-y_0=6(x-x_0)\end{align*} Plug in the point (200, 2500): y2500=6(x200)\begin{align*}y-2500=6(x-200)\end{align*} To find Anne’s base salary, we plug in x=0\begin{align*}x = 0\end{align*} and get y2500=1200y=1300\begin{align*}y-2500=-1200 \Rightarrow y=\ 1300\end{align*}.

Anne’s monthly base salary is $1300. #### Solving Real-World Problems Using Linear Models in Standard Form Here are two examples of real-world problems where the standard form of an equation is useful. #### Real-World Application: Buying Fruit Nadia buys fruit at her local farmer’s market. This Saturday, oranges cost$2 per pound and cherries cost $3 per pound. She has$12 to spend on fruit. Write an equation in standard form that describes this situation. If she buys 4 pounds of oranges, how many pounds of cherries can she buy?

Let’s define our variables:

xy=pounds of oranges=pounds of cherries\begin{align*}x &= \text{pounds of oranges}\!\\ y &= \text{pounds of cherries}\end{align*}

The equation that describes this situation is 2x+3y=12\begin{align*}2x+3y=12\end{align*}.

If she buys 4 pounds of oranges, we can plug x=4\begin{align*}x = 4\end{align*} into the equation and solve for y\begin{align*}y\end{align*}:

2(4)+3y=123y=1283y=4y=43\begin{align*}2(4)+3y=12 \Rightarrow 3y=12-8 \Rightarrow 3y=4 \Rightarrow y=\frac{4}{3}\end{align*}

Nadia can buy \begin{align*}1 \frac{1}{3}\end{align*} pounds of cherries.

### Example

#### Example 1

Peter skateboards part of the way to school and walks the rest of the way. He can skateboard at 7 miles per hour and he can walk at 3 miles per hour. The distance to school is 6 miles. Write an equation in standard form that describes this situation. If he skateboards for \begin{align*}\frac{1}{2}\end{align*} an hour, how long does he need to walk to get to school?

Let’s define our variables:

\begin{align*}x &= \text{time Peter skateboards}\!\\ y &= \text{time Peter walks}\end{align*}

The equation that describes this situation is: \begin{align*}7x+3y=6\end{align*}

If Peter skateboards \begin{align*}\frac{1}{2}\end{align*} an hour, we can plug \begin{align*}x = 0.5\end{align*} into the equation and solve for \begin{align*}y\end{align*}:

\begin{align*}7(0.5)+3y=6 \Rightarrow 3y=6-3.5 \Rightarrow 3y=2.5 \Rightarrow y=\frac{5}{6}\end{align*}

Peter must walk \begin{align*}\frac{5}{6}\end{align*} of an hour.

### Review

For 1-8, write the equation in slope-intercept, point-slope and standard forms.

1. The line has a slope of \begin{align*}\frac{2}{3}\end{align*} and contains the point \begin{align*}\left(\frac{1}{2}, 1 \right)\end{align*}.
2. The line has a slope of -1 and contains the point \begin{align*}\left(\frac{4}{5}, 0 \right)\end{align*}.
3. The line has a slope of 2 and contains the point \begin{align*}\left(\frac{1}{3}, 10 \right)\end{align*}.
4. The line contains points (2, 6) and (5, 0).
5. The line contains points (5, -2) and (8, 4).
6. The line contains points (-2, -3) and (-5, 1).

For 9-10, solve the problem.

1. Andrew has two part time jobs. One pays $6 per hour and the other pays$10 per hour. He wants to make $366 per week. Write an equation in standard form that describes this situation. If he is only allowed to work 15 hours per week at the$10 per hour job, how many hours does he need to work per week in his $6 per hour job in order to achieve his goal? 2. Anne invests money in two accounts. One account returns 5% annual interest and the other returns 7% annual interest. In order not to incur a tax penalty, she can make no more than$400 in interest per year. Write an equation in standard form that describes how much she should invest to earn the maximum interest without penalty. If she invests \$5000 in the 5% interest account, how much money can she invest in the other account?

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