Word Problems: Linear Models

Word problems are helpful for understanding why we are learning certain mathematical operations. They show real-life situations where we can put our math knowledge to work. Sometimes word problems ask us to write a linear function to model a particular situation.

Word problems using the slope-intercept form

Some word problems are phrased in such a way that we can easily find a linear function using the slope-intercept form of the equation for a line.

Remember, the slope-intercept form is in the form of:

$y = m x + b$

Example 1

Elaine's electric 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 Elaine uses zero electricity, that is when $x = 0$ , the bill is $15.00. Therefore, 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 the slope-intercept form $y = m x + b$

$y = 0.11 x + 15$

Word problems using the point-slope form

Some word problems are phrased in such a way that we can easily find a linear function using the point-slope form of the equation for a line.

Remember, the formula for the point-slope form of a line is:

$y - y_{1} = m (x - x_{1})$

Example 2

Anthony lives in Telluride, Colorado, but his job is located in Denver, Colorado. 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, his odometer shows that he has traveled 240,218 miles since he bought the car.

Write a linear model that gives the odometer reading of the car as a function of the number of weeks since Anthony started his new job.

First, find the rate of change. Be sure to multiply the distance by 2 to find his round-trip distance since he has to go and come back.

$2 (332) = 664$ 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.

That's okay, though. We have the coordinates of a point $(20, 240218)$ . So we can use the point-slope form $y - y_{1} = m (x - x_{1})$ .

$y - 240218 = 664 (x - 20)$

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

$y = 664 x + 226938$

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