example

The pitch of a building’s roof is the slope of the roof. Knowing the pitch is important in climates where there is heavy snowfall. If the roof is too flat, the weight of the snow may cause it to collapse. What is the slope of the roof shown?

Answer: Solution

Use the slope formula.	[latex]m=\Large\frac{\text{rise}}{\text{run}}[/latex]
Substitute the values for rise and run.	[latex]m={\Large\frac{\text{9 ft}}{\text{18 ft}}}[/latex]
Simplify.	[latex]m={\Large\frac{1}{2}}[/latex]
The slope of the roof is [latex]{\Large\frac{1}{2}}[/latex]

example

Have you ever thought about the sewage pipes going from your house to the street? Their slope is an important factor in how they take waste away from your house. Sewage pipes must slope down [latex]{\Large\frac{1}{4}}[/latex] inch per foot in order to drain properly. What is the required slope?

Answer: Solution

Use the slope formula.	[latex]m=\Large\frac{\text{rise}}{\text{run}}[/latex]
[latex]m={\Large\frac{-\frac{1}{4}\text{in}\text{.}}{1\text{ft}}}[/latex]
[latex]m={\Large\frac{-\frac{1}{4}\text{in}\text{.}}{1\text{ft}}}[/latex]
Convert [latex]1[/latex] foot to [latex]12[/latex] inches.	[latex]m={\Large\frac{-\frac{1}{4}\text{in}\text{.}}{12\text{in.}}}[/latex]
Simplify.	[latex]m=-{\Large\frac{1}{48}}[/latex]
The slope of the pipe is [latex]-{\Large\frac{1}{48}}[/latex]

Example

Given the dataset, verify the values of the slopes of each equation. Linear equations describing the change in median home values between [latex]1950[/latex] and [latex]2000[/latex] in Mississippi and Hawaii are as follows: Hawaii:  [latex]y=3966x+74,400[/latex] Mississippi:  [latex]y=924x+25,200[/latex] The equations are based on the following dataset. [latex]x[/latex] = the number of years since [latex]1950[/latex], and y = the median value of a house in the given state.

Year (x)	Mississippi House Value (y)	Hawaii House Value (y)
[latex]0[/latex]	[latex]$25,200[/latex]	[latex]$74,400[/latex]
[latex]50[/latex]	[latex]$71,400[/latex]	[latex]$272,700[/latex]

The slopes of each equation can be calculated with the formula you learned in the section on slope.

[latex] \displaystyle m=\frac{{{y}_{2}}-{{y}_{1}}}{{{x}_{2}}-{{x}_{1}}}[/latex]

Mississippi:

Name	Ordered Pair	Coordinates
Point 1	[latex](0, 25200)[/latex]	[latex]\begin{array}{l}x_{1}=0\\y_{1}=25200\end{array}[/latex]
Point 2	[latex](50, 71400)[/latex]	[latex]\begin{array}{l}x_{2}=50\\y_{2}=71400\end{array}[/latex]

[latex] \displaystyle m=\frac{{71,400}-{25,200}}{{50}-{0}}=\frac{{46,200}}{{50}} = 924[/latex]

We have verified that the slope [latex] \displaystyle m = 924[/latex] matches the dataset provided.   Hawaii:

Name	Ordered Pair	Coordinates
Point 1	[latex](0, 74400)[/latex]	[latex]\begin{array}{l}x_{1}=1950\\y_{1}=74400\end{array}[/latex]
Point 2	[latex](50, 272700)[/latex]	[latex]\begin{array}{l}x_{2}=2000\\y_{2}=272700\end{array}[/latex]

[latex]\displaystyle m=\frac{{272,700}-{74,400}}{{50}-{0}}=\frac{{198,300}}{{50}} = 3966[/latex]

We have verified that the slope [latex] \displaystyle m = 3966[/latex] matches the dataset provided.

Example

Given the dataset, verify the values of the slopes of the equation. A linear equation describing the change in the number of high school students who smoke, in a group of [latex]100[/latex], between [latex]2011[/latex] and [latex]2015[/latex] is given as:

 [latex]y = -1.75x+16[/latex]

And is based on the data from this table, provided by the Centers for Disease Control. [latex]x[/latex] = the number of years since [latex]2011[/latex], and [latex]y[/latex] = the number of high school smokers per [latex]100[/latex] students.

Year	Number of  High School Students Smoking Cigarettes (per 100)
[latex]0[/latex]	[latex]16[/latex]
[latex]4[/latex]	[latex]9[/latex]

Name	Ordered Pair	Coordinates
Point 1	[latex](0, 16)[/latex]	[latex]\begin{array}{l}x_{1}=0\\y_{1}=16\end{array}[/latex]
Point 2	[latex](4, 9)[/latex]	[latex]\begin{array}{l}x_{2}=4\\y_{2}=9\end{array}[/latex]

[latex] \displaystyle m=\frac{{{y}_{2}}-{{y}_{1}}}{{{x}_{2}}-{{x}_{1}}}=\frac{{9-16}}{{4-0}} =\frac{{-7}}{{4}}=-1.75[/latex]

We have verified that the slope [latex] \displaystyle{m=-1.75}[/latex] matches the dataset provided.

Example

Interpret the slope of each equation for house values using words. Hawaii:  [latex]y = 3966x+74,400[/latex] Mississippi:  [latex]y = 924x+25,200[/latex]

Answer: It helps to apply the units to the points that we used to define slope.  The x-values represent years, and the y-values represent dollar amounts. For Mississippi:

[latex] \displaystyle m=\frac{{71,400}-{25,200}}{{0}-{50}}=\frac{{46,200\text{ dollars}}}{{50\text{ year}}} = 924\frac{\text{dollars}}{\text{year}}[/latex]

Answer

The slope for the Mississippi home prices equation is positive, so each year the price of a home in Mississippi increases by [latex]924[/latex] dollars. We can apply the same thinking for Hawaii home prices. The slope for the Hawaii home prices equation tells us that each year, the price of a home increases by [latex]3966[/latex] dollars.

Example

Interpret the slope of the line describing the change in the number of high school smokers using words. Apply units to the formula for slope. The [latex]x[/latex] values represent years, and the y values represent the number of smokers. Remember that this dataset is per [latex]100[/latex] high school students.

[latex] \displaystyle m=\frac{{9-16}}{{2015-2011}} =\frac{{-7 \text{ smokers}}}{{4\text{ year}}}=-1.75\frac{\text{ smokers}}{\text{ year}}[/latex]

The slope of this linear equation is negative, so this tells us that there is a decrease in the number of high school age smokers each year. The number of high schoolers that smoke decreases by [latex]1.75[/latex] per [latex]100[/latex] each year.