# 5.3: Applications Using Linear Models

**At Grade**Created by: CK-12

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**Practice**Applications Using Linear Models

What if your car rental company charges $25 per day plus $0.25 per mile? When the car is returned to you the trip odometer reads 324 miles and the customer's bill totals $156. How could you determine the number of days the customer rented the car? In this Concept, you'll be able to solve real-world problems like this one.

### Watch This

CK-12 Foundation: 0503S Solving Real-World Problems with Linear Equations (H264)

### Guidance

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

#### Example A

*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?*

**Solution**

Let’s define our variables:

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

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 dollars per mile, so the truck company charges 50 cents per mile ($0.5 = 50 cents). Plugging in the slope and the

To find out the cost of driving the truck 220 miles, we plug in

**Driving 220 miles would cost $150.**

#### Example B

*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?*

**Solution**

Let’s define our variables:

We see that we 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 the point is (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

**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.

#### Example C

*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?*

**Solution**

Let’s define our variables:

The equation that describes this situation is

If she buys 4 pounds of oranges, we can plug

**Nadia can buy 113 pounds of cherries.**

Watch this video for help with the Examples above.

CK-12 Foundation: Solving Real-World Problems with Linear Equations

### Vocabulary

- A common form of a line (linear equation) is
**slope-intercept form:**y=mx+b , wherem is the slope and the point(0,b) is they− intercept .

- Often, we don’t know the value of the
y− intercept, but we know the value ofy for a non-zero value ofx . In this case, it’s often easier to write an equation of the line in**point-slope form.**An equation in point-slope form is written asy−y0=m(x−x0) , wherem is the slope and(x0,y0) is a point on the line.

- An equation in
**standard form**is writtenax+by=c , wherea,b , andc are all integers anda is positive. (Note that theb in the standard form is different than theb in the slope-intercept form.)

### Guided Practice

*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 12 an hour, how long does he need to walk to get to school?*

**Solution**

Let’s define our variables:

The equation that describes this situation is:

If Peter skateboards

**Peter must walk 56 of an hour.**

### Practice

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

- The line has a slope of
23 and contains the point(12,1) . - The line has a slope of -1 and contains the point
(45,0) . - The line has a slope of 2 and contains the point
(13,10) . - 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 this problem. If she invests $5000 in the 5% interest account, how much money does she need to invest in the other account?

### Image Attributions

## Description

## Learning Objectives

Here you'll learn how to solve real-world problems whose equations are straight lines in either point-slope, slope-intercept, or standard form.