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5.8: Slope of a Line Using Two Points

Difficulty Level: At Grade Created by: CK-12
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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*}ΔyΔx.

 \begin{align*}\frac{\Delta y}{\Delta x}=\frac{{y}_{2}-{y}_{1}}{{x}_{2}-{x}_{1}}\end{align*}ΔyΔx=y2y1x2x1

 \begin{align*}{y}_{2}-{y}_{1}\end{align*}y2y1represents the vertical change, or difference, between two points on a line.

 \begin{align*}{x}_{2}-{x}_{1}\end{align*}x2x1represents 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*}ΔyΔx 

Next, substitute in the given information.

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

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

Then, slope = \begin{align*}\frac{7}{4}\end{align*}74.

The answer is \begin{align*}\frac{7}{4}\end{align*}74 .

Example 2

Find the slope of line \begin{align*}CD\end{align*}CD 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*}ΔyΔx=y2y1x2x1 

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*}yy1x2x1=\begin{align*}\frac{9-3}{9-7}\end{align*}9397

Then, simplify.

 \begin{align*}\frac{6}{2}=3\end{align*}62=3

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*}FG below.

 

First, write the formula.

  \begin{align*}\frac{\Delta y}{\Delta x}=\frac{{y}_{2}-{y}_{1}}{{x}_{2}-{x}_{1}}\end{align*}ΔyΔx=y2y1x2x1

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*}7336 \begin{align*}\frac{{y}_{}-{y}_{1}}{{x}_{2}-{x}_{1}}=\end{align*}yy1x2x1= 

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

 \begin{align*}\frac{4}{-3}=-\frac{4}{3}\end{align*}43=43 

\begin{align*}-\frac{4}{3}\end{align*}43The 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*}37. 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*}ΔyΔx    

Then substitute the values that are given.

 \begin{align*}\frac{3}{7}=\end{align*}37= \begin{align*}\frac{\Delta y}{\Delta x}\end{align*}ΔyΔx

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

The change in x, \begin{align*}\Delta x\end{align*}Δx, 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*}23. 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*}ΔyΔx

Then substitute the values that are given.

  \begin{align*}-\frac{2}{3}\end{align*}23   = \begin{align*}\frac{\Delta y}{\Delta x}\end{align*}ΔyΔx

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

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

The change in x, \begin{align*}\Delta x\end{align*}Δx, 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*}12.

  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

 

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Difficulty Level:
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Aug 10, 2015
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Sep 08, 2016
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