### Applications using Linear Models

Solve word problems using the equation of a straight line.

#### 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:

Peter pays a flat fee of $40 for the day; this is the -intercept.

He pays $63 for 46 miles; this is the coordinate point (46,63).

Start with the point-slope form of the line:

Plug in the coordinate point:

Plug in the point (0, 40):

Solve for the slope:

The slope is 0.5, so the truck company charges 0.5 dollars, or 50 cents, per mile ($0.5 = 50 cents). Plugging in the slope and the -intercept, the equation of the line is .

To find out the cost of driving the truck 220 miles, we substitute 220 for to get .

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. Use the equation to determine Anne’s monthly base salary.

First define the variables:

You are given the slope and a point on the line:

Nadia gets $6 for each shade, so the slope is 6.

She made $2500 when she sold 200 shades, so there is a point at (200, 2500).

Start with the point-slope form of the line:

Plug in the slope:

Plug in the point (200, 2500):

To find Anne’s base salary, we plug in and get

Anne’s monthly base salary is $1300.

#### 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:

The equation that describes this situation is

If she buys 4 pounds of oranges, we can plug into the equation and solve for :

Nadia can buy 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 an hour, how long does he need to walk to get to school?

Define the variables:

The equation that describes this situation is:

Peter skateboards an hour, so substitute into the equation and solve for

Peter must walk of an hour to get to school.

### Review

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

- The line has a slope of and contains the point
- The line has a slope of -1 and contains the point
- The line has a slope of 2 and contains the point
- The line contains points (2, 6) and (5, 0).
- The line contains points (5, -2) and (8, 4).
- The line contains points (-2, -3) and (-5, 1).

For 9-10, solve the problem.

- 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?
- 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?

### Review (Answers)

To view the Review answers, open this PDF file and look for section 5.3.