The document provides examples of using slope in word problems to model real-world scenarios using equations. It demonstrates how to write equations to represent total cost, height, or earnings as a function of other variables like time, distance, or hours. Examples include writing equations to model taxi fare as a function of miles, a candle's height burning over time, and a babysitter's earnings based on hours worked. The document emphasizes using the slope-intercept form, y=mx+b, to set up equations that can then be used to calculate values for other variables.

1. Skill31D Using Slope in Word Problems or Applications of Slope
This is the final lesson in a series on slope. We will concentrate on how the
slope-intercept form of a line can give us information about a “word problem.”
Remember that the equation y = mx + b is called the slope intercept form and
m is the slope or rate of change while b is the y-intercept or beginning value or
sometimes referred to as the flat fee in a word problem.
Maria is traveling by taxi while on
business in Chicago. The taxi charges a
flat fee of $4 and then $2 per mile.
a. Write an equation to represent the
total cost, y, of a taxi ride of x miles.
b. How much would a taxi ride of
eight miles cost?
c. Rewrite the equation you wrote in
part a but let the total cost be
represented by c and the number
miles be represented by n.
We will solve this problem as an
example on the next slides.
Notice that there are several parts.
You cannot do part b or c without doing
Part a first.

2. Maria is traveling by taxi while on business in Chicago. The taxi charges
a flat fee of $4 and then $2 per mile.
a. Write an equation to represent the total cost, y, of a taxi ride of x
miles.
b. How much would a taxi ride of eight miles cost?
c. Rewrite the equation you wrote in part a but let the total cost be
represented by c and the number miles be represented by n.
a. Y = 2x + 4 Why would we write this? The $4 is the flat fee. That is what you
would pay if you just got in the taxi and rode 0 miles. The rate per mile, $2, is
the slope. Rates are slopes. So m = 2 and b = 4. Thus in terms of x and y, we
would write y = 2x, + 4.
b. If you took the taxi for 8 hours, you would pay the flat fee, $4, and 8(2) or $16
for the 8 miles. This would be $20. This is what the slope intercept form of the
line gives when you let x = 8 in Y = 2x + 4. So you can just use the equation
you write on part a and let x = 8. Y = 2(8) + 4. That is y = 16 + 4 or y = 20.
c. The total cost is the y in y = 2x + 4. The miles are represented by x. So, we like to
use letters that come from the words they represent. So write c = 2n + 4. This is the
same thing as y = mx + b but more like the situation.

3. a. Y = 2x + 4
b. Y = 2(8) + 4
Y = 16 + 4
Y = 20
c. C = 2n + 4
Here is the work for the last problem.

4. An electrician charges $80 for a house call.
In addition, he charges $65 per hour for labor.
a. Write an equation to model the
total cost, c, of a repair job of h hours.
b. If the repair job took three hours to
complete, how much would the total cost be?
c. If the total cost of an electrical repair
job was $210, find the number of hours it
took.
Now we will work on this problem. See the next slide for part a.

5. An electrician charges $80 for a house call. In addition, he
charges $65 per hour for labor.
a. Write an equation to model the total cost, c, of a
repair job of h hours.
Notice that the problem mentions a “flat fee” of $80 and a per hour charge.
This is a “red flag” that the problem is a “y = mx + b” type. Note that not all word
Problems are like this. This one is!
So start by writing out y = mx + b. The directions say to let the total cost be c.
Do you see that in y = mx + b, the y is the result of adding mx and b. This is a totaling.
The job last h hours. This is the x. M is the rate. Here that is $65. The flat fee is $80.
So write c = 65 h + 80 That would be how much it would cost you to hire the
electrician for h hours.
C = 65 h + 80 Important---this could
also be written as C = 80 + 65h. Do you see how
that is really the same thing.

6. Now look at part b. Use c = 65h + 80 to see how much you would pay for 3 hours or
In other words let h = 3.
b. If the repair job took three hours
to complete, how much would the
total cost be?
C = 65(3) + 80
C = 195 + 80
C = 275
Now that we know the equation for the relationship between the cost and the hours,
we can find either one if we know the other. In part c. we know the cost is $210, so how
many hours did the plumber work? Use your equation solving skills to figure it out.
c. C = 65 h + 80
210 = 65 h + 80
-80 -80
130 = 65 h
130 = 65 h
65 65
2 = h

7. A new candle is eight inches tall. It burns at a rate of 0.5 inches per hour.
a. Write an equation to model the height of the candle, h, after t hours.
b. How tall will the candle be after burning for five hours?
a. H = -0.5 t + 8 I started with y = mx + b and let
the total height be H while x is the number of
hours, t.
b. H = -0.5(5) + 8 This means let t = 5 and see
what you get. Notice the candle is burning so
the slope is negative.
H = -2.5 + 8
H = 5.5

8. Here are some examples. Try them. Answers are on following slides.
1. To join a local gym, there is a $50 joining fee and then $20 per
month. Write an equation using C for the total cost of joining the
gym for n months. How much would the total cost be for 6
months?
2. The water level on the Locust Fork River is 35 feet and it is
dropping at a rate of 0.5 feet per day. Write and equation for the
water level, L, after d days. What will be the water level after 6
days? In how many days will the level be 20 feet?
3. Sheila’s daughter babysits for $5 per hour plus a flat fee of $6.
Write an equation for the money she would earn, K, if she
worked h hours. How much would she earn in 7 hours. How
many hours would she have to work to earn $51?

9. 1. C = 20 N + 50 For 6 months, let N = 6 and get C = 170 dollars.
2. L = -0.5 d + 35 After 6 days, L = -0.5(6) + 35 or 32 feet.
It would take 30 days for the level to get to 20 feet because
20 = -0.5 d + 35 solves to give d = 30.
3. K = 5h + 6 She would earn $41 for 7 hours since K = 5(7) + 6 IS 41.
The last question needs to solve 51 = 5h + 6 or h = 9 hous.