Linear Models

Models are extremely useful tools for analyzing relationships and making predictions based on the those relationships. Linear models allow us to represent many situations that involve constant rate of change.

Building Linear Models

The steps of building a linear model include...

1. Identify changing quantities and define variables to represent these quantities.
2. Identify information that provide values for the variables or parts of a functional model like slope and initial value.
3. Determine what we are trying to find, identify, solve, or interpret.
4. Identify a way to arrive at the solution we are trying to find.
5. Write a formula for the function.
6. Solve or evaluate the function.
7. Reflect on whether your answer is reasonable for the given situation.
8. Convey the result using appropriate units.

Let's use this in an example...

$\underline{Example}$

A company sells doughnuts. They incur a fixed cost of $\$25,000$ for rent, insurance, and other expenses. It costs $\$0.25$ to produce each doughnut.

(a) Write a linear model to represent the cost $C$ of the company as a function of $x$, the number of doughnuts produced.

(b) Find and interpret the $y$-intercept.

The changing quantity is the number of doughnuts produced which we express as $x$. It costs $\$0.25$ to produce each doughnut. There is also a fixed cost of $\$25,000$ that the company occurs. This cost is the initial value. Finally, we are trying to find the total cost ($C$) which we can arrive to by adding the two costs.

$C = 25000 + 0.25x$ can be used to represent this relationship. This is in the form $y = mx + b$ where $b = 25000$ in this case. This means the $y$-intercept is $(0, 25000)$.

We can use this model $C = 25000 + 0.25x$ to arrive at our total cost in dollars. If we sell zero doughnuts, then our cost is $\$25000$.

Systems of Linear Equations

Given a situation that represents a system of linear equations, we can solve them by...

1. Identify the input and output of each linear model.
2. Create each linear model using the information aquired.
3. Find the solution by setting the two linear functions equal to another and solve for $x$ which is the point of intersection on a graph.

An example of this is...

$\underline{Example}$

Jamal is choosing between two truck-rental companies. The first, Keep on Trucking, Inc., charges an up-front fee of $\$20$, then $59$ cents a mile. The second, Move it Your Way, charges an up-front fee of $\$16$, then $63$ cents a mile. When will Keep on Trucking, Inc. be a better choice for Jamal?

Let $x$ be the number of miles driven. Keep on Trucking, Inc.'s cost can be modeled with $C = 0.59x + 20$ and Move it Your Way's cost can be modeled with $C = 0.63x + 16$.

$0.63x + 16 = 0.59x + 20$

$0.63x = 0.59x + 4$

$0.04x = 4$

$x = 100$

It takes at least $100$ miles to make Keep on Trucking, Inc. a better choice for Jamal.