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# 5.7: Slope

Difficulty Level: Basic Created by: CK-12
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Casey's family is trying new things on their adventure vacation. Casey is excited about riding a zipline for the very first time. The beginning point is 20 meters higher than the end point, and the length of the zipline is 200 meters. What is the slope of the line?

In this concept, you will learn about slopes and how to graph them.

### Slope

A slope can be defined as the steepness of a straight line. It is the ratio of the change in vertical measure to the change in horizontal measure. This is also referred to as the ratio of "rise" to  "run"  or the change in y to the change in x.

The Greek letter Delta, Δ\begin{align*}\Delta \end{align*}is used to show the changes in x and y.

Δy\begin{align*}\Delta y\end{align*} stands for the change in y.

Δx\begin{align*}\Delta x\end{align*} stands for the change in x.

Line AB on the coordinate plane below has a change of 5 units vertically, Δy\begin{align*}\Delta y\end{align*}

The slope of line AB is the ratio riserun\begin{align*}\frac{rise}{run}\end{align*}or  ΔyΔx\begin{align*}\frac{\Delta y}{\Delta x}\end{align*} = 56\begin{align*}\frac{5}{6}\end{align*} .

Line AB, with a slope of 56\begin{align*}\frac{5}{6}\end{align*}, is a positive slope.The line slants upward from left to right.

Slopes can be positive or negative. Regardless of the quadrant, or point values, slopes are based on the change of y (increase or decrease) as the x value moves from left to right.

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

A line that slants down from left to right has a negative slope.

Horizontal and vertical lines also have a slope. A horizontal line has a change in x, but it does not have a change in y.

slope = ΔyΔx\begin{align*}\frac{\Delta y}{\Delta x}\end{align*} = 0Δx\begin{align*}\frac{0}{\Delta x}\end{align*} = 0

The slope of a horizontal line is 0.

A vertical line has a change in y, but it does not have a change in x.

slope = ΔyΔx\begin{align*}\frac{\Delta y}{\Delta x}\end{align*} = Δy0\begin{align*}\frac{\Delta y}{0}\end{align*}= undefined

Division by zero is undefined.

The slope of a vertical line is undefined.

### Examples

#### Example 1

Earlier, you were given a problem about Casey and the zip line.

The change in elevation, vertical distance, was 20m, and the change in horizontal distance was 200m. What is the slope?

First, write the equation.

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

Next, substitute in the given values.

ΔyΔx=20200\begin{align*}\frac{\Delta y}{\Delta x}=\frac{20}{200}\end{align*}

Then, reduce.

The answer is that the slope equals 110\begin{align*}\frac{1}{10}\end{align*}

#### Example 2

Determine if the slope of the line below is positive or negative. What is its value?

First, determine if the line rises or falls as it moves from left to right.

The line falls as it moves from left to right. The slope is negative.

Next, use the equation to find the value of the slope.

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

total change in y = -8 units

total change in x = 6 units

ΔyΔx=-86\begin{align*}\frac{\Delta y}{\Delta x}=\frac{\text{-}8}{6}\end{align*}

Then, reduce to lowest terms.

-86=-1.33\begin{align*}\frac{\text{-}8}{6}=\text{-}1.33\end{align*}

Next, remember that the slope is negative.

The answer is the slope equals 1.33\begin{align*}-1.33\end{align*}

#### Example 3

Determine the slope:

First, determine if the slope is positive or negative.

The line rises as it moves from left to right. The slope is positive.

Next, use the formula.

slope =  ΔyΔx\begin{align*}\frac{\Delta y}{\Delta x}\end{align*} = 27\begin{align*}\frac{2}{7}\end{align*}

The answer is the slope equals 27\begin{align*}\frac{2}{7}\end{align*}

Example 4

Identify the slope of each line shown below.

1.

First, consider the line in a\begin{align*}a\end{align*}.

The line is vertical.

Next, remember that a vertical line has a slope that is undefined.

The answer is that the slope of line a is undefined.

Then, consider the line in b\begin{align*}b\end{align*}.

The line is horizontal.

Next, remember that a horizontal line has a slope equal to zero.

The answer is that the slope of line b equals 0.

#### Example 5

Murray broke his arm and could not use the rope to climb into his treehouse. Instead, he got a ladder from the garage and set it up against the structure. The treehouse is 15 feet off the ground, and Murray set it 3 feet away from the side. What is the slope of the ladder? Is it positive or negative?

First write the formula.

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

Next, substitute in what is known.

ΔyΔx=153\begin{align*}\frac{\Delta y}{\Delta x}=\frac{15}{3}\end{align*}

Then, reduce to lowest terms.

The answer is that the slope equals 5.

Whether the slope is positive or negative can only be determined if there is a diagram or graphing coordinates given.

### Review

For each graph, tell if the slope of the line shown is positive, negative, zero, or undefined.

1. Does a positive slope have to contain positive numbers?
2. True or false. A horizontal line is undefined.
3. True or false. A negative slope goes down from right to left.
4. True or false. A vertical line has an undefined slope.
5. True or false. You can figure out any slope as long as the line has some slant to it.

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

Color Highlighted Text Notes

### Vocabulary Language: English

Horizontally

Horizontally means written across in rows.

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 $m$

Vertically

Vertically means written up and down in columns.

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