Applications of Systems of Linear Equations

Renting a car

You are going on a road trip and want to rent a car. You need to decide which agency to rent a car from. Agency A charges \$50 plus 10 cents (0.10 dollars) for each mile you drive. Agency B charges \$30 plus \$0.20 per mile. Agency C charges \$0.50 for each mile you drive. The cost in dollars for renting a car for one day from these three different rental agencies and driving it $d$ miles is given by the following equations:

$A=50+0.10d$
$B=30+0.20d$
$C=0.50d$

Graphs of the cost equation for each of the three agencies are shown on the grid to the left.

If you drive 150 miles, how much will each agency charge you? (Click or tap somewhere on the $d=150$ gridline, and the computer will calculate the answer for you on the right hand side of the grid.) Agency A: \$
Agency B: \$
Agency C: \$
If you are only planning on driving 80 miles, which agency will have the cheapest car rental?
How many miles do you have to drive for the cost of renting from agency A and agency C to be the same?
When is it cheapest to rent from agency B? Between and miles
If you are planning on driving 400 miles, which agency will have the cheapest car rental?

Cost, revenue, and profit

A company that manufactures and sells running shoes has a fixed overhead cost of \$65000 — that is, they need to spend \$65000 to start making shoes, no matter how many they make. It costs the company an additional \$20 to produce each pair of running shoes and it sells each pair of shoes for \$70.

Write an equation to describe $C$, the total cost to the company in terms of $n$, the number of pairs of shoes it produces. (It costs the company \$65020 to produce one pair of shoes, \$65040 to produce 2 pairs of shoes, \$65060 to produce 3 pairs of shoes, etc.)
Write an equation that describes the total revenue, $R$, in terms of $n$, the number of pairs of shoes the company sells. (The revenue of the company is its income from selling shoes.)

Slide the values of $a$ and $b$ so that the equations for $C=65000+an$ and $R=bn$ match the equations you found in findShoeEqs. Click or tap inside the grid and drag or use the left and right arrow keys to answer the following questions.

What is the cost of making 500 pairs of shoes?
What is the revenue if the company sells 500 pairs of shoes?
Does the company make or lose money if it sells 500 pairs of shoes?
How much money does the company make if it sells 500 pairs of shoes? (If the company loses money, your answer should be negative.)

The amount of money a company makes (its revenue minus its cost) is called its profit.

How many pairs of shoes should the company sell in order for the cost and revenue to be the same?
What is the company’s profit if it sells 2000 pairs of shoes?
Slide the $b$ slider to 150. What would the equation for $R$ be if the company sold each pair of shoes for \$150?
If the company sold each pair of shoes for \$150, how many pairs of shoes would it have to sell before it could make a profit?
What would the equation for revenue be if the company sold each pair of shoes for \$20?
Use the slider for $b$ to change the equation for $R$ to the equation you found. Are the two lines parallel?
Which line is always higher, the purple line or the green line?
Would the company ever be able to make a profit at this price?

Which cell phone plan should you use?

Supertel Communications offers two different cell phone plans: the Platinum Plan and the Gold Plan. On the Platinum Plan you pay a \$30 monthly fee and 10¢ (\$0.10) for every minute that you talk on the phone. The Gold Plan has no monthly fee, but you pay 30¢ (\$0.30) for every minute of use. If $t$ is the number of minutes you speak on the phone in a month, $P$ the monthly cost in dollars of using the Platinum Plan, and $G$ the monthly cost in dollars of using the Gold Plan, write equations relating $P$ and $G$ to $t$.
Platinum Plan:
Gold Plan:

Slide the values of $a$, $b$, and $c$ so that the equations $P=a+bt$ and $G=ct$ match the equations you found in findPhoneEqs.

If you expect to talk on the phone for 200 minutes this month, will the Platinum or the Gold plan be cheapest?
Which plan will be cheapest if you expect to talk on the phone for 75 minutes?
At what value of $t$ do the graphs of $P$ and $G$ intersect?
How many minutes would you need to talk in order for the two plans to be equally expensive?

Birth and death rates

The tables below show the world birth and death rates per 1000 people for the years 1990-2015. The data from these tables is plotted on the grid to the left.

Birth Rate
Death Rate

There are two lines which you will move to try to find a good “fit” to the data from the tables. Look to the right of the grid. The numbers 3.047 and 1.733 are called the root-mean-square errors of the lines. The root-mean-square error tells you how far away a line is from the data (a smaller number means the line is a better “fit” to the data).

On the grid to the left, the equations $B=mt+n$ (for the birth rate) and $D=pt+q$ (for the death rate) are graphed. Alternate sliding the values of $m$ and $n$, as well as $p$ and $q$, until you find approximately the best equation for each set of data. Your root-mean-square error should be less than 0.7 for $B$ and less than 0.1 for $D$. (On the vertical axis you see the rate of birth and death per 1000 people. On the horizontal axis you see the number of years since 1990.)

What equation did you find for $B$, the birth rate? (The equations for the two lines are written below the grid.)
What equation did you find for $D$, the death rate?
If the lines continue to be a good approximation for the birth and death rates in each year, about when will the world birth and death rates be the same? (Remember that $t$ is the number of years since 1990.)
If this occurs, will the population of the world increase or decrease after this year?