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(Video) Slope-Intercept Word Problems

by cpfaffinator

This video by cpfaffinator explains how to translate word problems into variables while working through a couple of examples.

(Note: the answer at 3:34 should be $y=0.50x+3$ instead of $y=0.5x + b$.)

$\large{\text{Summary}}$

When writing linear equations (also called linear models) from word problems, you need to know what the $x$ and $y$ variables refer to, as well as what the slope and $y$-intercept are. Here are some tricks to help translate:

1. The $x$ variable. Usually, you'll be asked to find an equation in terms of some factor. This will usually be whatever affects the output of the equation.
2. The $y$ variable. The $y$ variable is usually whatever you're trying to find in the problem. It'll depend on the $x$ variable.
3. The slope ($m$). The slope will be some sort of rate or other change over time.
4. The $y$-intercept ($b$). Look for a starting point or extra fees—something that will be added (or subtracted) on, no matter what the $x$ variable is.

Using these parts, we can write the word problem in slope intercept form, $y=mx+b$, and solve.

Example 1

A city parking garage charges a flat rate of $\$3.00$ for parking $2$ hours or less, and $\$0.50$ per hour for each additional hour. Write a linear model that gives the total charge in terms of additional hours parked.

First up, we want the equation, or model, to give us the total charge. This will be what the $y$ variable represents. $$y=\text{total charge for parking}$$ The total charge depends on how many additional hours you've parked. This is the $x$ variables. $$x=\text{additional hours parked}$$

Next, we need to find the slope, $m$. If we look at the slope-intercept equation, $y=mx+b$, the slope is multiplied with the input $x$. So, we want to find some rate in our problem that goes with the number of additional hours parked. In the problem, we have $\$0.50$ per hour for each additional hour. The number of hours is being multiplied by the price of $50\text{ cents}$ every hour. $$m=0.50$$

The $y$-intercept is the starting point. In other words, if we parked our car in the garage for $0$ hours, what would the price be? The problem says the garage *charges a flat rate of $\$3.00$. A flat rate means that no matter what we're being charged $3\text{ dollars}$. $$b=3$$

Plugging these into the slope-intercept equation will give us our model. $$\begin{align} y&=mx+b\\ y&=0.50x+3 \end{align}$$

Example 2

Six $2\text{ foot}$ tall pine trees were planted during the school's observation of Earth Awareness Week in $1990$. The trees have grown at an average rate of $\frac34\text{ foot per year}$. Write a linear model that gives the height of the trees in terms of the number of years since they were planted.

Since the trees grow a certain amount every year, the height of the trees depends on how many years it has been since the trees were planted. The $y$ variable depends on the $x$ variable. $$y=\text{ height of the tree}$$

The $x$ variable will affect the $y$ variable. So, the height of the trees is affected by how many years they've been growing. $$x=\text{ years since the trees were planted}$$

Slope is the rate or change over time. The trees literally change, or grow, every year. So the rate is $\frac34\text{ feet per year}$. $$m=\frac34$$

The $y$-intercept will be the $y$ value at the start, or when $x=0$. So it's asking what is the height of the tree at the start, or when $0\text{ years}$ have gone by? When the trees were first planted, they were $2\text{ feet}$ tall. $$b=2$$

Plugging in $m=\frac34$ and $b=2$ into the slope-intercept equation gives us: $$\begin{align} y&=mx+b\\ y&=\frac34x+2 \end{align}$$

Translate to a Linear Equation

Sometimes, the trick to solving a word problem will be to translate it into a linear equation. To clue you in, linear equation word problems usually involve some sort of rate of change, or steady increase (or decrease) based on a single variable. If you see the word rate, or even "per" or "each", it's a safe bet that a word problem is calling for a linear equation.

There are a couple steps when translating from a word problem to a linear equation. Review them, then we'll work through an example.

1. Find the $y$ variable, or output. What is the thing you're trying to find? This will often be a price, or an amount of time, or something else countable that depends on other things.
2. Find the $x$ variable, or input. What is affecting the price, or amount of time, etc.?
3. Find the slope. What's the rate in the problem?
4. Find the $y$-intercept. Is there anything that's added or subtracted on top of the rate, no matter what what $x$ is?
5. Plug all the numbers you know into $y = mx + b$!

Example

Wally and Cobb are starting a catering business. They rent a kitchen for $\$350$ a month, and charge $\$75$ for each event they cater. If they cater $12$ events in a month, how much do they profit?

Right, the problem asks how much Wally and Cobb profit. That means that their profit is the output we want from our equation—our $y$ variable.

To find the $x$ variable, look for words like "rate," "per," "each," or something else that implies change over time or a rate of change.

In this case, we can see that Wally and Cobb charge "$\$75$ for each event they cater"—which means the money they make in a month will have something to do with the number of events they cater. Therefore, this will be our $x$ variable!

Since they'll make $\$75$ for each event they cater, $\$75$ is their rate! That means it's our slope—our $m$ in $y = mx + b$. $$m=75$$

Each month, Wally and Cobb have to pay rent! That subtracts $\$350$ from their profits, so their $y$-intercept will always be $-350$.

Now all that's left to do is put all our information into an equation. We know that: $$\begin{align} m &= 75\\ b &= -350 \end{align}$$

Plugging this into the slope-intercept equation gives us: $$\begin{align} y&=mx+b\\ y &= 75x - 350 \end{align}$$

But we're not quite done yet. Let's look at the word problem again to refresh our memories.

Wally and Cobb are starting a catering business. They rent a kitchen for $\$350$ a month, and charge $\$75$ for each event they cater. If they cater $12$ events in a month, how much do they profit?

Solution

We want to solve for $y$ when $x=12$. $$\begin{align} y&=75x-350\\ y &= 75({\color{#8925ae}{12}}) - 350\\ y &= 900 - 350\\ y &= 550 \end{align}$$ If they cater $12$ events in a month, Wally and Cobb will make $\$550$ in profit.

Word Problems with Linear Relationships

When you're faced with a linear word problem, it can feel daunting to work out a solution. The good news is that there are few simple steps that you can take to tackle linear word problems.

For future linear word problems, you can use these same steps to tackle the problem!