# Word Problems: Linear Models

Word problems sometimes ask us to write a linear function to model a situation.

The word problem may be phrased in such a way that we can easily find a linear function using the slope-intercept form of the equation for a line.

Example 1:

Hannah's electricity company charges her $0.11$ per kWh (kilowatt-hour) of electricity, plus a basic connection charge of $15.00$ per month. Write a linear function that models her monthly electricity bill as a function of electricity usage.

Here, when Hannah uses zero electricity (that is, when $x=0$), the bill is $15.00$. So, the $y$-intercept is $15$.

The rate of change is $0.11$. That is, for each increase of $x$ by $1$ unit (in this case, kilowatt-hours), there is an increase in $y$ by $0.11$.

Substitute in the slope-intercept form $y=mx+b$.

$y=0.11x+15$

In other problems, it may be easier to use the point-slope form of the equation.

Example 2:

Roger has a house in Telluride, Colorado, but starts a new job in Denver. Every Monday, he drives his car $332$ miles from Telluride to Denver, spends the week in a company apartment, and then drives back to Telluride on Friday. He doesn't use his car for anything else.

After $20$ weeks of this, his odometer shows that he has travelled $240,218$ miles since he bought the car.

Write a linear model which gives the odometer reading of the car as a function of the number of weeks since Roger started the new job.

First, find the rate of change. Be sure to multiply the distance by $2$: he has to go and come back!

$2\left(332\right)=664$ miles per week

This represents the slope of the line. But, since we are not given the odometer reading of the car before he starts the job, we don't know the $y$-intercept yet.

But that's okay. We have the coordinates of a point: $\left(20,240218\right)$. So we can use the point-slope form $y-{y}_{1}=m\left(x-{x}_{1}\right)$.

$y-240218=664\left(x-20\right)$

Note that you can use this equation to find the $y$-intercept if you want to.