<img src="https://d5nxst8fruw4z.cloudfront.net/atrk.gif?account=iA1Pi1a8Dy00ym" style="display:none" height="1" width="1" alt="" />
Dismiss
Skip Navigation

5.8: Slope of a Line Using Two Points

Difficulty Level: At Grade Created by: CK-12
Atoms Practice
Estimated16 minsto complete
%
Progress
Practice Slope of a Line Using Two Points
Practice
Progress
Estimated16 minsto complete
%
Practice Now
Turn In

Hadiya has a very active dog that she would like to train for agility. Hadiya's dad is helping her make a ramp off their back deck so Hadiya and Max can practice. Measuring from the ground at the base of the deck, the ramp will be placed out 4 feet, the horizontal distance, and reach the 7-foot-high deck, the vertical distance. Using the base of the deck as point  (0,0), solve for the slope of the agility ramp.

In this concept, you will learn to solve for the slope of a line using two points on a line, or coordinates.

Solving for Slope of a Line Using Coordinates

Coordinates are values on a line that show a point's exact position. They are indicated by parentheses, (x,y). The point (7,-4) means that the x value of a particular point is 7 units to the right of point (0,0) and the y value of the same point is 4 units below point (0,0).

If the graph is given, the slope of a line can be found by choosing two points on the line and counting the units vertically and horizontally to find the change in y over the change in x, \begin{align*}\frac{\Delta y}{\Delta x}\end{align*}.

 \begin{align*}\frac{\Delta y}{\Delta x}=\frac{{y}_{2}-{y}_{1}}{{x}_{2}-{x}_{1}}\end{align*}

 \begin{align*}{y}_{2}-{y}_{1}\end{align*}represents the vertical change, or difference, between two points on a line.

 \begin{align*}{x}_{2}-{x}_{1}\end{align*}represents the horizontal change, or difference, between two points on a line.

Since slope is a ratio, improper fractions are not to be converted to mixed numbers.

Examples

Example 1

Earlier, you were given a problem about Hadiya and her agile dog, Max.

Hadiya's dad is making an agility ramp that rises 7 feet and stretches out 4 feet from a point (0,0) at the base of the deck in their backyard. What is the slope of the ramp?

First, write the equation.

slope = \begin{align*}\frac{\Delta y}{\Delta x}\end{align*} 

Next, substitute in the given information.

The change in y, \begin{align*}\Delta y\end{align*} , goes from 0 to 7 feet, a change of +7.

The change in x,  \begin{align*}\Delta x\end{align*}, goes from 0 to 4 feet. Assume to the right and a change of +4.

Then, slope = \begin{align*}\frac{7}{4}\end{align*}.

The answer is \begin{align*}\frac{7}{4}\end{align*} .

Example 2

Find the slope of line \begin{align*}CD\end{align*} below.

First, write the formula for the slope.

   \begin{align*}\frac{\Delta y}{\Delta x}=\frac{{y}_{2}-{y}_{1}}{{x}_{2}-{x}_{1}}\end{align*} 

Next, fill in the values that are known from the coordinates given, using C as Point 1 and D as Point 2.

 \begin{align*}\frac{{y}_{}-{y}_{1}}{{x}_{2}-{x}_{1}}=\end{align*}\begin{align*}\frac{9-3}{9-7}\end{align*}

Then, simplify.

 \begin{align*}\frac{6}{2}=3\end{align*}

The answer is that the slope of line CD equals 3. The slope is positive.

Example 3

Find the slope of line \begin{align*}FG\end{align*} below.

 

First, write the formula.

  \begin{align*}\frac{\Delta y}{\Delta x}=\frac{{y}_{2}-{y}_{1}}{{x}_{2}-{x}_{1}}\end{align*}

Next, fill in the values that are known from the coordinates given using F as Point 1 and G as Point 2.

 \begin{align*}\frac{7-3}{3-6}\end{align*} \begin{align*}\frac{{y}_{}-{y}_{1}}{{x}_{2}-{x}_{1}}=\end{align*} 

Then, simplify. Remember that slopes are not given as mixed numbers.

 \begin{align*}\frac{4}{-3}=-\frac{4}{3}\end{align*} 

\begin{align*}-\frac{4}{3}\end{align*}The answer is that the slope of line FG equals 

The slope is negative

Example 4

Draw a line that goes through a point at (–4, –1) and has a slope of \begin{align*}\frac{3}{7}\end{align*}. What are the coordinates of the second point?

First, on a coordinate plane, plot the given point at (-4, -1)

Next, remember the formula for the slope of a line.

slope =   \begin{align*}\frac{\Delta y}{\Delta x}\end{align*}    

Then substitute the values that are given.

 \begin{align*}\frac{3}{7}=\end{align*} \begin{align*}\frac{\Delta y}{\Delta x}\end{align*}

This means that the change in y, \begin{align*}\Delta y\end{align*}, is equal to +3.

The change in x, \begin{align*}\Delta x\end{align*}, is equal to +7.

Next, starting at the given point (-4, -1), move up +3 units on the y-axis and right +7 units on the x-axis.

Then, mark the point, read its location, and draw a line through both points.

The answer is (3, 2).

Example 5

Draw a line that goes passes through (–5, 4) and has a slope of \begin{align*}- \frac{2}{3}\end{align*}. Give the coordinates of a second point.

First, on a coordinate plane, plot the given point at (-5, 4).

Next, remember the formula for the slope of a line.

slope =   \begin{align*}\frac{\Delta y}{\Delta x}\end{align*}

Then substitute the values that are given.

  \begin{align*}-\frac{2}{3}\end{align*}   = \begin{align*}\frac{\Delta y}{\Delta x}\end{align*}

Since the slope is negative, the minus sign can be assigned to either value.

The change in y, \begin{align*}\Delta y\end{align*}, is equal to -2.

The change in x, \begin{align*}\Delta x\end{align*}, is equal to +3.

Next, starting at the given point (-5, 4), move down 2 units on the y-axis and right 3 units on the x-axis.Then, mark the point, read its location, and draw a line through both points.

The answer is (-2, 2).

There are an infinite number of points along this line, one of which could be identified by solving the above with the negative assigned to the change in x rather than the change in y.

Review

Find the slope of each line shown.

  1. On the coordinate grid below, draw a line that passes through (–3, 2) and has a slope of \begin{align*}\frac{1}{2}\end{align*}.

  1. Is this slope positive or negative?
  2. On the coordinate grid below, draw a line that passes through (-2, 5) and has a slope of -4.

  1. Is this slope positive or negative?

Review (Answers)

To see the Review answers, open this PDF file and look for section 5.8.

Resources

 

Notes/Highlights Having trouble? Report an issue.

Color Highlighted Text Notes
Show More

Image Attributions

Show Hide Details
Description
Difficulty Level:
At Grade
Grades:
Date Created:
Aug 10, 2015
Last Modified:
Aug 11, 2016
Save or share your relevant files like activites, homework and worksheet.
To add resources, you must be the owner of the Modality. Click Customize to make your own copy.
Please wait...
Please wait...
Image Detail
Sizes: Medium | Original
 
Here