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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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Remember the book count from the last Concept? Well, now let's take it one step further. Take a look.

Five of the students in Mrs. Henderson’s class have been tracking the number of books that they have read and have been comparing their results. During the first week, all five finished one book. After the second week, all five had finished two books. During the third week, all five had finished three books. After ten weeks, all five had finished ten books.

Here is a graph of the students' results.

Can you figure out the slope of this line?

This Concept is about finding the slope of a line. By the end of it, you will know how to answer this question.

### Guidance

Suppose you are given the graph of a line. You can find the slope of that line by choosing two points on the line. Then you can count the units to find the ratio of the rise to the run.

Be sure to consider if the line slants up or down from left to right. This will help you determine if the slope is positive or negative.

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

From point \begin{align*}C\end{align*} to point \begin{align*}D\end{align*}, the line rises 6 units up and then runs 2 units to the right. Since the lines slants up from left to right, the slope is positive.

\begin{align*}\text{Slope} = \frac{rise}{run} = \frac{6}{2} = \frac{6 \div 2}{2 \div 2} = \frac{3}{1} = 3.\end{align*}

The slope of line \begin{align*}CD\end{align*} is 3.

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

From point \begin{align*}F\end{align*} to point \begin{align*}G\end{align*}, the rise is 4 units down and then 3 units to the right. Since the line slants down from left to right, the slope is negative. You can think of a rise of 4 units down as a negative rise, and represent it as -4.

\begin{align*}\text{Slope} = \frac{rise}{run} = \frac{-4}{3} = - \frac{4}{3}.\end{align*}

The slope of line \begin{align*}FG\end{align*} is \begin{align*}- \frac{4}{3}\end{align*}.

Remember, the slope represents a ratio, not an improper fraction. We cannot say that the slope of line \begin{align*}FG\end{align*} is \begin{align*}-1 \frac{1}{3}\end{align*}.

We know that we can think of the slope of a line as the ratio \begin{align*}\frac{rise}{run}\end{align*}. We can use this ratio to draw a line with a specific slope.

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

Here are the steps to drawing a line with a given slope.

On a coordinate plane, plot a point at (–4, –1). Place the tip of your pencil at that point.

The slope is \begin{align*}\frac{3}{7}\end{align*}. So, \begin{align*}\text{slope} = \frac{rise}{run} = \frac{3}{7}\end{align*}.

The slope is positive, so move your pencil 3 units up and then 7 units to the right to find another point on that same line. Your pencil will end up at (3, 2), so plot a point there. Then draw a line through the two points.

The line you drew passes through a point at (–4, –1) and has a slope of \begin{align*}\frac{3}{7}\end{align*}.

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

On a coordinate plane, plot a point at (–5, 4). Place the tip of your pencil at that point.

The slope is \begin{align*}-\frac{2}{3}\end{align*}. So, \begin{align*}\text{slope} = \frac{rise}{run} = \frac{-2}{3}\end{align*}.

The slope is negative, so move your pencil 2 units down and then 3 units to the right to find another point on that same line. Your pencil will end up at (-2, 2), so plot a point there. Then draw a line through the two points.

The line you drew passes through a point at (–5, –4) and has a slope of \begin{align*}-\frac{2}{3}\end{align*}.

Now it's time for you to try a few on your own. Identify the slope of each line.

#### Example A

Solution: Slope = \begin{align*}\frac{2}{7}\end{align*}

#### Example B

True or false. A line that slants down from left to right has a positive slope.

Solution: False

#### Example C

True or false. A line that slants up from left to right has a positive slope.

Solution: True

Here is the original problem once again.

Five of the students in Mrs. Henderson’s class have been tracking the number of books that they have read and have been comparing their results. During the first week, all five finished one book. After the second week, all five had finished two books. During the third week, all five had finished three books. After ten weeks, all five had finished ten books.

Here is a graph of the students' results.

Can you figure out the slope of this line?

You can see that the rate of change is 0. The slope is 0. The line is a horizontal line because the students did not increase or decrease the number of books that the read per week. The numbers remained consistent.

### Vocabulary

Here are the vocabulary words in this Concept.

Slope
the slant of a line or the steepness of a line. It is represented on a graph by a ratio of rise over run.
Rise
the vertical measurement of a line.
Run
the horizontal measurement of a line.
Positive Slope
a slope that goes up from left to right.
Negative Slope
a slope that goes down from right to left.

### Guided Practice

Here is one for you to try on your own.

Define this slope.

Answer

This line is vertical. It goes up, but doesn't have a particular rise or run that you can calculate.

The slope is undefined.

### Video Review

Here is a video for review.

### Practice

Directions: Find the slope of each line shown.

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

12. 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*}.

13. Is this slope positive or negative?

14. On the coordinate grid below, draw a line that passes through (-2, 5) and has a slope of -4.

15. Is this slope positive or negative?

### Notes/Highlights Having trouble? Report an issue.

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### Vocabulary Language: English

Negative Slope

A line with a negative slope will slant down from left to right.

Positive Slope

A line with a positive slope will slant up from left to right.

Rise

When calculating slope, the rise is the vertical distance between two specified points.

Run

When calculating slope, the run is the horizontal distance traveled.

Slope

Slope is a measure of the steepness of a line. A line can have positive, negative, zero (horizontal), or undefined (vertical) slope. The slope of a line can be found by calculating “rise over run” or “the change in the $y$ over the change in the $x$.” The symbol for slope is $m$

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Nov 30, 2012
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