Models with One Line

Section 9.1 Models with One Line

The key to word problems is that we want to use the mathematical tools we've already developed. We just need to turn the sentences into points and equations.

You might have an instict for word problems. If that is the case, great, feel free to jump to the checkpoints below. However, many students are very uncomfortable with word problems. The hard part is translating the sentences into equations. Here is a step-by-step procedure that you might find helpful:

Fact 9.1. Solving Linear Word Problems.

To solve a word problem about linear functions, follow the following steps:

Write the information as a list. Our brains have trouble seeing the important information in a paragraph, so writing it as a short list makes it easier to see what you're working with.
Look at your units and decide which one is for your $x$-values and which one is for your $y$-values. Some of this decision will come with practice, but here are a few general guilding principles:
- Time is almost always an $x$-value
- Money is usually a $y$-value
Using what you figured out in step 2, determine what kind of information you have on your list from step 1. There are three kinds of information you might be given:
- The slope (always a rate)
- A point (an $x$ and $y$ that go together)
- The $y$-intercept (a $y$-value that occurs when $x=0$)
Use the information above to write an equation for a line, just like we did in Section 7.2.
Use the equation from step 4 to answer the overall question.

Let's see this strategy in an example:

Example 9.2.

Here is the problem: A taxi ride costs a flat fee of $3 and $1.50 per mile. How much does a 12-mile ride cost?

Let's go through the steps:

We have the following information:
- flat fee of $3
- rate of $1.50 per mile
- Question: cost of 12-mile ride
We have two units: miles and dollars. Using the guiding principles above, we are going to use dollars as the $y$-values, which leaves miles as the $x$-values.
Let's categorize the information from the first step:
- flat fee of $3: This is a cost ($y$-value) that happens before you drive any miles (when $x=0$), so this is the $y$-intercept.
- rate of $1.50 per mile: this is a rate, which means it's our slope.
- Question: cost of 12-mile ride: since cost is the $y$-values and miles are the $x$-values, we need to figure out what $y$ is when $x=12$
In the last step, we figured out that we have a slope of 1.5 and a $y$-intercept of 3, so our equation is $f(x)=1.5x+3\text{,}$ or $y=1.5x+3\text{.}$
To answer the overall question, we will plug in $x=12\text{:}$

\begin{equation*} y = 1.5x+3 = 1.5(12)+3 = 18+3 = 21 \end{equation*}

Therefore, we have our final answer: a 12-mile ride costs $21.

Checkpoint 9.3.

A gym membership has a $13 registration fee and then a fixed price per month. After 12 months, you had paid a total $97. How long have you been a member at the gym if you have paid $146?

Answer.

19 months

Solution.

Let's go through the steps:

We have the following information:
- $13 registration fee
- after 12 months, paid $97
- Question: length of membership with cost $146
We have two units: months and dollars. Using the guiding principles above, we are going to use dollars as the $y$-values and time (months) as the $x$-values.
Let's categorize the information from the first step:
- $13 registration fee: This is a cost ($y$-value) that happens before any time has passed (when $x=0$), so this is the $y$-intercept.
- after 12 months, paid $97: This is a time ($x$-value) and a cost ($y$-value) that go together, which is a point.
- Question: length of membership with cost $146: since cost is the $y$-values and time is the $x$-value, we need to figure out what $x$ is when $y=146$
In the last step, we figured out that we have a $y$-intercept of 13 and a point $(12,97)\text{.}$ We can use this information to find the slope, since the $y$-intercept gives us a point of $(0,3)\text{:}$

\begin{equation*} \frac{97-13}{12-0} = \frac{84}{12}=7 \end{equation*}

Now we can write our equation: $f(x)=7x+13\text{,}$ or $y=7x+13\text{.}$
To answer the overall question, we will plug in $y=146\text{:}$

\begin{align*} 146\amp= 7x+13\\ 133\amp = 7x \\ 19 \amp = x \end{align*}

Therefore, we have our final answer: you have paid $146 after 19 months.