Suppose a movie rental service charges a fixed fee per month and also charges $3.00 per movie rented. Last month you rented 8 movies and your monthly bill was $30.00. Could you write a linear equation to model this situation? Would slope-intercept form, point-slope form, or standard form be easiest to use?

### Applying Linear Models

Modeling linear relationships can help solve real-world applications. Consider the example situations below, and note how different problem-solving methods may be used in each.

- Nadia has $200 in her savings account. She gets a job that pays $7.50 per hour and she deposits all her earnings in her savings account. Write the equation describing this problem in slope-intercept form. How many hours would Nadia need to work to have $500 in her account?

Begin by defining the variables:

\begin{align*}y=\end{align*} amount of money in Nadia’s savings account

\begin{align*}x=\end{align*} number of hours

The \begin{align*}y\end{align*}-intercept ($200) and the slope of the equation \begin{align*}\left( \tfrac{\$7.50}{1 \text{ hour}} \right)\end{align*} are given.

We are told that Nadia has $200 in her savings account, so \begin{align*}b=200\end{align*}.

We are told that Nadia has a job that pays $7.50 per hour, so \begin{align*}m=7.50\end{align*}.

By substituting these values into slope–intercept form, \begin{align*}y=mx+b\end{align*}, we obtain \begin{align*}y=7.5x+200.\end{align*}

To answer the question, substitute $500 for the value of \begin{align*}y\end{align*} and solve.

\begin{align*}500 &= 7.5x+200 \\ 7.5x &=300\\ x &=40\end{align*}

Nadia must work 40 hours if she is to have $500 in her account.

- Marciel rented a moving truck for the day. Marciel remembers only that the rental truck company charges $40 per day and some amount 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?

Define the variables: \begin{align*}x=\end{align*} distance in miles; \begin{align*}y=\end{align*} cost of the rental truck in dollars. There are two ordered pairs: (0, 40) and (46, 63).

**Step 1:** Begin by finding the slope: \begin{align*}\tfrac{63-40}{46-0}=\tfrac{23}{46}=\tfrac{1}{2}\end{align*}

**Step 2:** Substitute the slope for \begin{align*}m\end{align*} and one of the coordinates for \begin{align*}(x_1,y_1)\end{align*}.

\begin{align*}y-40= \frac{1}{2} (x-0)\end{align*}

To find out how much will it cost to rent the truck for 220 miles, substitute 220 for the variable \begin{align*}x\end{align*}.

\begin{align*}y-40 & = \frac{1}{2} (220-0)\\ y-40 & =0.5(220)\\ y & =\$150\end{align*}

- Nimitha 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?

Define the variables: \begin{align*}x=\end{align*} pounds of oranges and \begin{align*}y=\end{align*} pounds of cherries.

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

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

\begin{align*}2(4)+3y &=12 \\ 3y &=12-8 \\ 3y &=4 \\ y &=\frac{4}{3}\end{align*}

Nimitha can buy \begin{align*}1\tfrac{1}{3}\end{align*} pounds of cherries.

### Examples

#### Example 1

Earlier, you were told that a movie rental service charges a fixed fee per month and also charges $3.00 per movie rented. Last month you rented 8 movies and your monthly bill was $30.00. What linear equation would model this situation?

In this example, you are given the slope of the line that would represent this situation: 3 (because each rental costs $3.00). You are also given the point (8, 30) because when you rent 8 movies, your bill is $30.00. So, you have the slope and a point. This means that the best form to use to write an equation is point-slope form.

To write the equation, first define the variables: \begin{align*}x=\end{align*} number of movies rented; \begin{align*}y=\end{align*} the monthly bill in dollars. The slope is 3 and one ordered pair is (8, 30).

Since you have the slope, substitute the slope for \begin{align*}m\end{align*} and the coordinate for \begin{align*}(x_1,y_1)\end{align*} into the point-slope form equation:

\begin{align*}y-30= 3 (x-8)\end{align*}

You can rewrite this in slope-intercept form by using the Distributive Property and the Addition Property of Equality:

\begin{align*}y-30 &= 3 (x-8)\\ y-30 &= 3x-24\\ y &= 3x + 6\end{align*}

So the equation that models this situation is \begin{align*}y-30= 3 (x-8)\end{align*} or \begin{align*}y = 3x+6\end{align*}.

#### Example 2

A stalk of bamboo of the family *Phyllostachys nigra* grows at steady rate of 12 inches per day and achieves its full height of 720 inches in 60 days. Write the equation describing this problem in slope-intercept form. How tall is the bamboo 12 days after it started growing?

Define the variables.

\begin{align*}y=\end{align*} the height of the bamboo plant in inches

\begin{align*}x=\end{align*} number of days

The problem gives the slope of the equation and a point on the line.

The bamboo grows at a rate of 12 inches per day, so \begin{align*}m=12\end{align*}.

We are told that the plant grows to 720 inches in 60 days, so we have the point (60, 720).

\begin{align*}\text{Start with the slope-intercept form of the line.} && y& =mx+b\\ \text{Substitute 12 for the slope.} && y& =12x+b\\ \text{Substitute the point} \ (60,720). && 720& =12(60)+b \Rightarrow b=0\\ \text{Substitute the value of} \ b \ \text{back into the equation.} && y& =12x\end{align*}

To answer the question, substitute the value \begin{align*}x=12\end{align*} to obtain \begin{align*}y=12(12)=144\end{align*} inches.

The bamboo is 144 inches 12 days after it starts growing.

#### Example 3

Jethro skateboards part of the way to school and walks for 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 Jethro skateboards for \begin{align*}\tfrac{1}{2}\end{align*} of an hour, how long does he need to walk to get to school?

Define the variables: \begin{align*}x=\end{align*} hours Jethro skateboards and \begin{align*}y=\end{align*} hours Jethro walks.

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

If Jethro skateboards \begin{align*}\tfrac{1}{2}\end{align*} of an hour, we substitute \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 \\ 3y &=6 - 3.5 \\ 3y &=2.5\\ y &=\frac{5}{6}\end{align*}

Jethro must walk \begin{align*}\tfrac{5}{6}\end{align*} of an hour to get to school.

### Review

- To buy a car, Andrew puts in a down payment of $1500 and pays $350 per month in installments. Write an equation describing this problem in slope-intercept form. How much money has Andrew paid at the end of one year?
- Anne transplants a rose seedling in her garden. She wants to track the growth of the rose, so she measures its height every week. In the third week, she finds that the rose is 10 inches tall and in the eleventh week she finds that the rose is 14 inches tall. Assuming the rose grows linearly with time, write an equation describing this problem in slope-intercept form. What was the height of the rose when Anne planted it?
- Ravi hangs from a giant exercise spring whose length is 5 m. When his child Nimi hangs from the spring, his length is 2 m. Ravi weighs 160 lbs. and Nimi weighs 40 lbs. Write the equation for this problem in slope-intercept form. What should we expect the length of the spring to be when his wife Amardeep, who weighs 140 lbs., hangs from it?
- Petra is testing a bungee cord. She ties one end of the bungee cord to the top of a bridge and to the other end she ties different weights. She then measures how far the bungee stretches. She finds that for a weight of 100 lbs., the bungee stretches to 265 feet and for a weight of 120 lbs., the bungee stretches to 275 feet. Physics tells us that in a certain range of values, including the ones given here, the amount of stretch is a linear function of the weight. Write the equation describing this problem in slope-intercept form. What should we expect the stretched length of the cord to be for a weight of 150 lbs?
- Nadia is placing different weights on a spring and measuring the length of the stretched spring. She finds that for a 100 gram weight the length of the stretched spring is 20 cm and for a 300 gram weight the length of the stretched spring is 25 cm. Write an equation in point-slope form that describes this situation. What is the unstretched length of the spring?
- Andrew is a submarine commander. He decides to surface his submarine to periscope depth. It takes him 20 minutes to get from a depth of 400 feet to a depth of 50 feet. Write an equation in point-slope form that describes this situation. What was the submarine’s depth five minutes after it started surfacing?
- 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 her 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?
- The farmer’s market sells tomatoes and corn. Tomatoes are selling for $1.29 per pound and corn is selling for $3.25 per pound. If you buy 6 pounds of tomatoes, how many pounds of corn can you buy if your total spending cash is $11.61?
- The local church is hosting a Friday night fish fry for Lent. They sell a fried fish dinner for $7.50 and a baked fish dinner for $8.25. The church sold 130 fried fish dinners and took in $2,336.25. How many baked fish dinners were sold?
- 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 at 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?

### Review (Answers)

To see the Review answers, open this PDF file and look for section 5.6.