example

Graph the line passing through the point [latex]\left(1,-1\right)[/latex] whose slope is [latex]m=\frac{3}{4}[/latex]. Solution Plot the given point, [latex]\left(1,-1\right)[/latex]. Use the slope formula [latex]m=\frac{\text{rise}}{\text{run}}[/latex] to identify the rise and the run. [latex-display]\begin{array}{}\\ \\ m=\frac{3}{4}\hfill \\ \frac{\text{rise}}{\text{run}}=\frac{3}{4}\hfill \\ \\ \\ \text{rise}=3\hfill \\ \text{run}=4\hfill \end{array}[/latex-display] Starting at the point we plotted, count out the rise and run to mark the second point. We count [latex]3[/latex] units up and [latex]4[/latex] units right. Then we connect the points with a line and draw arrows at the ends to show it continues. We can check our line by starting at any point and counting up [latex]3[/latex] and to the right [latex]4[/latex]. We should get to another point on the line.

example

Graph the line with [latex]y[/latex] -intercept [latex]\left(0,2\right)[/latex] and slope [latex]m=-\frac{2}{3}[/latex].

Answer: Solution Plot the given point, the [latex]y[/latex] -intercept [latex]\left(0,2\right)[/latex]. Use the slope formula [latex]m=\frac{\text{rise}}{\text{run}}[/latex] to identify the rise and the run. [latex-display]\begin{array}{}\\ \\ m=-\frac{2}{3}\hfill \\ \frac{\text{rise}}{\text{run}}=\frac{-2}{3}\hfill \\ \\ \\ \text{rise}=-2\hfill \\ \text{run}=3\hfill \end{array}[/latex-display] Starting at [latex]\left(0,2\right)[/latex], count the rise and the run and mark the second point. Connect the points with a line.

example

Graph the line passing through the point [latex]\left(-1,-3\right)[/latex] whose slope is [latex]m=4[/latex].

Answer: Solution Plot the given point.

Identify the rise and the run.	[latex]m=4[/latex]
Write [latex]4[/latex] as a fraction.	[latex]\frac{\text{rise}}{\text{run}}=\frac{4}{1}[/latex]
	[latex]\text{rise}=4\text{run}=1[/latex]

Count the rise and run. Mark the second point. Connect the two points with a line.

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=\frac{\text{rise}}{\text{run}}[/latex]
Substitute the values for rise and run.	[latex]m=\frac{\text{9 ft}}{\text{18 ft}}[/latex]
Simplify.	[latex]m=\frac{1}{2}[/latex]
The slope of the roof is [latex]\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]\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=\frac{\text{rise}}{\text{run}}[/latex]
[latex]m=\frac{-\frac{1}{4}\text{in}\text{.}}{1\text{ft}}[/latex]
[latex]m=\frac{-\frac{1}{4}\text{in}\text{.}}{1\text{ft}}[/latex]
Convert [latex]1[/latex] foot to [latex]12[/latex] inches.	[latex]m=\frac{-\frac{1}{4}\text{in}\text{.}}{12\text{in.}}[/latex]
Simplify.	[latex]m=-\frac{1}{48}[/latex]
The slope of the pipe is [latex]-\frac{1}{48}[/latex].