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### Be Prepared 11.10

Before you get started, take this readiness quiz.

Simplify: $\frac{1-4}{8-2}.$

If you missed this problem, review Example 4.49.

### Be Prepared 11.11

Divide: $\frac{0}{4},\frac{4}{0}.$

If you missed this problem, review Example 7.37.

### Be Prepared 11.12

Simplify: $\frac{15}{\mathrm{-3}},\frac{\mathrm{-15}}{3},\frac{\mathrm{-15}}{\mathrm{-3}}.$

If you missed this problem, review Example 4.47.

As we’ve been graphing linear equations, we’ve seen that some lines slant up as they go from left to right and some lines slant down. Some lines are very steep and some lines are flatter. What determines whether a line slants up or down, and if its slant is steep or flat?

The steepness of the slant of a line is called the slope of the line. The concept of slope has many applications in the real world. The pitch of a roof and the grade of a highway or wheelchair ramp are just some examples in which you literally see slopes. And when you ride a bicycle, you feel the slope as you pump uphill or coast downhill.

### Use Geoboards to Model Slope

In this section, we will explore the concepts of slope.

Using rubber bands on a geoboard gives a concrete way to model lines on a coordinate grid. By stretching a rubber band between two pegs on a geoboard, we can discover how to find the slope of a line. And when you ride a bicycle, you __feel__ the slope as you pump uphill or coast downhill.

### Manipulative Mathematics

Doing the Manipulative Mathematics activity "Exploring Slope" will help you develop a better understanding of the slope of a line.

We’ll start by stretching a rubber band between two pegs to make a line as shown in Figure 11.17.

Does it look like a line?

Now we stretch one part of the rubber band straight up from the left peg and around a third peg to make the sides of a right triangle as shown in Figure 11.18. We carefully make a $90\xb0$ angle around the third peg, so that one side is vertical and the other is horizontal.

To find the slope of the line, we measure the distance along the vertical and horizontal legs of the triangle. The vertical distance is called the *rise* and the horizontal distance is called the *run*, as shown in Figure 11.19.

To help remember the terms, it may help to think of the images shown in Figure 11.20.

On our geoboard, the rise is $2$ units because the rubber band goes up $2$ spaces on the vertical leg. See Figure 11.21.

What is the run? Be sure to count the spaces between the pegs rather than the pegs themselves! The rubber band goes across $3$ spaces on the horizontal leg, so the run is $3$ units.

The slope of a line is the ratio of the rise to the run. So the slope of our line is $\frac{2}{3}.$ In mathematics, the slope is always represented by the letter $m.$

### Slope of a line

The slope of a line is $m=\frac{\text{rise}}{\text{run}}.$

The rise measures the vertical change and the run measures the horizontal change.

What is the slope of the line on the geoboard in Figure 11.21?

$$\phantom{\rule{0ex}{0ex}}m=\frac{\text{rise}}{\text{run}}$$

$$m=\frac{2}{3}$$

$$\text{The line has slope}\phantom{\rule{0ex}{0ex}}\frac{2}{3}.$$

When we work with geoboards, it is a good idea to get in the habit of starting at a peg on the left and connecting to a peg to the right. Then we stretch the rubber band to form a right triangle.

If we start by going up the rise is positive, and if we stretch it down the rise is negative. We will count the run from left to right, just like you read this paragraph, so the run will be positive.

Since the slope formula has rise over run, it may be easier to always count out the rise first and then the run.

### Example 11.30

What is the slope of the line on the geoboard shown?

**Answer**Use the definition of slope.

$m=\frac{\text{rise}}{\text{run}}$

Start at the left peg and make a right triangle by stretching the rubber band up and to the right to reach the second peg.

Count the rise and the run as shown.

$\begin{array}{cccc}\text{The rise is}\phantom{\rule{0ex}{0ex}}3\phantom{\rule{0ex}{0ex}}\text{units}.\hfill & & & m=\frac{3}{\text{run}}\hfill \\ \text{The run is}\phantom{\rule{0ex}{0ex}}4\phantom{\rule{0ex}{0ex}}\text{units}.\hfill & & & m=\frac{3}{4}\hfill \\ & & & \text{The slope is}\phantom{\rule{0ex}{0ex}}\frac{3}{4}.\hfill \end{array}$

### Try It 11.58

What is the slope of the line on the geoboard shown?

### Try It 11.59

What is the slope of the line on the geoboard shown?

### Example 11.31

What is the slope of the line on the geoboard shown?

**Answer**Use the definition of slope.

$m=\frac{\text{rise}}{\text{run}}$

Start at the left peg and make a right triangle by stretching the rubber band to the peg on the right. This time we need to stretch the rubber band down to make the vertical leg, so the rise is negative.

$\begin{array}{cccc}\text{The rise is}\phantom{\rule{0ex}{0ex}}\mathrm{-1}.\hfill & & & m=\frac{\mathrm{-1}}{\text{run}}\hfill \\ \text{The run is}\phantom{\rule{0ex}{0ex}}3.\hfill & & & m=\frac{\mathrm{-1}}{3}\hfill \\ & & & m=-\frac{1}{3}\hfill \\ & & & \text{The slope is}\phantom{\rule{0ex}{0ex}}-\frac{1}{3}.\hfill \end{array}$

### Try It 11.60

What is the slope of the line on the geoboard?

### Try It 11.61

What is the slope of the line on the geoboard?

Notice that in the first example, the slope is positive and in the second example the slope is negative. Do you notice any difference in the two lines shown in Figure 11.22.

As you read from left to right, the line in Figure A, is going up; it has positive slope. The line Figure B is going down; it has negative slope.

### Example 11.32

Use a geoboard to model a line with slope $\frac{1}{2}.$

**Answer**To model a line with a specific slope on a geoboard, we need to know the rise and the run.

Use the slope formula. $m=\frac{\text{rise}}{\text{run}}$ Replace $m$ with $\frac{1}{2}$. $\frac{1}{2}=\frac{\text{rise}}{\text{run}}$ So, the rise is $1$ unit and the run is $2$ units.

Start at a peg in the lower left of the geoboard. Stretch the rubber band up $1$ unit, and then right $2$ units.

The hypotenuse of the right triangle formed by the rubber band represents a line with a slope of $\frac{1}{2}.$

### Try It 11.62

Use a geoboard to model a line with the given slope: $m=\frac{1}{3}.$

### Try It 11.63

Use a geoboard to model a line with the given slope: $m=\frac{3}{2}.$

### Example 11.33

Use a geoboard to model a line with slope $\frac{\mathrm{-1}}{4},$

**Answer**Use the slope formula. $m=\frac{\text{rise}}{\text{run}}$ Replace $m$ with $\mathrm{-}\frac{1}{4}$. $\mathrm{-}\frac{1}{4}=\frac{\text{rise}}{\text{run}}$ So, the rise is $\mathrm{-1}$ and the run is $4.$

Since the rise is negative, we choose a starting peg on the upper left that will give us room to count down. We stretch the rubber band down $1$ unit, then to the right $4$ units.

The hypotenuse of the right triangle formed by the rubber band represents a line whose slope is $-\frac{1}{4}.$

### Try It 11.64

Use a geoboard to model a line with the given slope: $m=\frac{\mathrm{-2}}{1}.$

### Try It 11.65

Use a geoboard to model a line with the given slope: $m=\frac{\mathrm{-1}}{3}.$

### Find the Slope of a Line from its Graph

Now we’ll look at some graphs on a coordinate grid to find their slopes. The method will be very similar to what we just modeled on our geoboards.

### Manipulative Mathematics

### Example 11.34

Find the slope of the line shown:

**Answer**Locate two points on the graph, choosing points whose coordinates are integers. We will use $(0,\mathrm{-3})$ and $(5,1).$

Starting with the point on the left, $(0,\mathrm{-3}),$ sketch a right triangle, going from the first point to the second point, $(5,1).$

Count the rise on the vertical leg of the triangle. The rise is 4 units. Count the run on the horizontal leg. The run is 5 units. Use the slope formula. $m=\frac{\text{rise}}{\text{run}}$ Substitute the values of the rise and run. $m=\frac{4}{5}$ The slope of the line is $\frac{4}{5}$. Notice that the slope is positive since the line slants upward from left to right.

### Try It 11.66

Find the slope of the line:

### Try It 11.67

Find the slope of the line:

### How To

#### Find the slope from a graph.

- Step 1. Locate two points on the line whose coordinates are integers.
- Step 2. Starting with the point on the left, sketch a right triangle, going from the first point to the second point.
- Step 3. Count the rise and the run on the legs of the triangle.
- Step 4. Take the ratio of rise to run to find the slope. $m=\frac{\text{rise}}{\text{run}}$

### Example 11.35

Find the slope of the line shown:

**Answer**Locate two points on the graph. Look for points with coordinates that are integers. We can choose any points, but we will use (0, 5) and (3, 3). Starting with the point on the left, sketch a right triangle, going from the first point to the second point.

(Video) 11.5 Point Slope Form of a Line - Video LessonCount the rise – it is negative. The rise is −2. Count the run. The run is 3. Use the slope formula. $m=\frac{\text{rise}}{\text{run}}$ Substitute the values of the rise and run. $m=\frac{\mathrm{-2}}{3}$ Simplify. $m=-\frac{2}{3}$ The slope of the line is $-\frac{2}{3}.$ Notice that the slope is negative since the line slants downward from left to right.

What if we had chosen different points? Let’s find the slope of the line again, this time using different points. We will use the points $(\mathrm{-3},7)$ and $(6,1).$

Starting at $(\mathrm{-3},7),$ sketch a right triangle to $(6,1).$

Count the rise. The rise is −6. Count the run. The run is 9. Use the slope formula. $m=\frac{\text{rise}}{\text{run}}$ Substitute the values of the rise and run. $m=\frac{\mathrm{-6}}{9}$ Simplify the fraction. $m=-\frac{2}{3}$ The slope of the line is $-\frac{2}{3}.$

It does not matter which points you use—the slope of the line is always the same. The slope of a line is constant!

### Try It 11.68

Find the slope of the line:

### Try It 11.69

Find the slope of the line:

The lines in the previous examples had $y$-intercepts with integer values, so it was convenient to use the *y*-intercept as one of the points we used to find the slope. In the next example, the $y$-intercept is a fraction. The calculations are easier if we use two points with integer coordinates.

### Example 11.36

Find the slope of the line shown:

**Answer**Locate two points on the graph whose coordinates are integers. $(2,3)$ and $(7,6)$ Which point is on the left? $(2,3)$ Starting at $(2,3)$, sketch a right angle to $(7,6)$ as shown below. Count the rise. The rise is 3. Count the run. The run is 5. Use the slope formula. $m=\frac{\text{rise}}{\text{run}}$ Substitute the values of the rise and run. $m=\frac{3}{5}$ The slope of the line is $\frac{3}{5}.$

### Try It 11.70

Find the slope of the line:

### Try It 11.71

Find the slope of the line:

### Find the Slope of Horizontal and Vertical Lines

Do you remember what was special about horizontal and vertical lines? Their equations had just one variable.

- horizontal line $y=b;$ all the $y$-coordinates are the same.
- vertical line $x=a;$ all the $x$-coordinates are the same.

So how do we find the slope of the horizontal line $y=4?$

What is the rise? | The rise is 0. |

What is the run? | The run is 3. |

What is the slope? | $m=\frac{\text{rise}}{\text{run}}$ |

$m=\frac{0}{3}$ | |

$m=0$ |

The slope of the horizontal line $y=4$ is $0.$

All horizontal lines have slope $0$. When the $y$-coordinates are the same, the rise is $0$.

### Slope of a Horizontal Line

The slope of a horizontal line, $y=b,$ is $0.$

Now we’ll consider a vertical line, such as the line $x=3$

What is the rise? | The rise is 2. |

What is the run? | The run is 0. |

What is the slope? | $m=\frac{\text{rise}}{\text{run}}$ |

$m=\frac{2}{0}$ |

But we can’t divide by $0.$ Division by $0$ is undefined. So we say that the slope of the vertical line $x=3$ is undefined. The slope of all vertical lines is undefined, because the run is $0.$

### Slope of a Vertical Line

The slope of a vertical line, $x=a,$ is undefined.

### Example 11.37

Find the slope of each line:

- ⓐ$\phantom{\rule{0ex}{0ex}}x=8\phantom{\rule{0ex}{0ex}}$
- ⓑ$\phantom{\rule{0ex}{0ex}}y=\mathrm{-5}$

**Answer**ⓐ $\phantom{\rule{0ex}{0ex}}x=8$

This is a vertical line, so its slope is undefined.

ⓑ $\phantom{\rule{0ex}{0ex}}y=\mathrm{-5}$

This is a horizontal line, so its slope is $0.$

### Try It 11.72

Find the slope of the line: $x=\mathrm{-4}.$

### Try It 11.73

Find the slope of the line: $y=7.$

### Quick Guide to the Slopes of Lines

### Use the Slope Formula to find the Slope of a Line between Two Points

Sometimes we need to find the slope of a line between two points and we might not have a graph to count out the rise and the run. We could plot the points on grid paper, then count out the rise and the run, but there is a way to find the slope without graphing.

Before we get to it, we need to introduce some new algebraic notation. We have seen that an ordered pair $(x,y)$ gives the coordinates of a point. But when we work with slopes, we use two points. How can the same symbol $(x,y)$ be used to represent two different points?

Mathematicians use subscripts to distinguish between the points. A subscript is a small number written to the right of, and a little lower than, a variable.

- $({x}_{1},{y}_{1})\phantom{\rule{0ex}{0ex}}\text{read}\phantom{\rule{0ex}{0ex}}x\phantom{\rule{0ex}{0ex}}\text{sub}\phantom{\rule{0ex}{0ex}}1,y\phantom{\rule{0ex}{0ex}}\text{sub}\phantom{\rule{0ex}{0ex}}1$
- $({x}_{2},{y}_{2})\phantom{\rule{0ex}{0ex}}\text{read}\phantom{\rule{0ex}{0ex}}x\phantom{\rule{0ex}{0ex}}\text{sub}\phantom{\rule{0ex}{0ex}}2,y\phantom{\rule{0ex}{0ex}}\text{sub}\phantom{\rule{0ex}{0ex}}2$

We will use $({x}_{1},{y}_{1})$ to identify the first point and $({x}_{2},{y}_{2})$ to identify the second point. If we had more than two points, we could use $({x}_{3},{y}_{3}),({x}_{4},{y}_{4}),$ and so on.

To see how the rise and run relate to the coordinates of the two points, let’s take another look at the slope of the line between the points $(2,3)$

Since we have two points, we will use subscript notation.

$$\stackrel{{x}_{1},{y}_{1}}{\left(2,3\right)}\phantom{\rule{0ex}{0ex}}\stackrel{{x}_{2},{y}_{2}}{\left(7,6\right)}$$

On the graph, we counted the rise of $3.$ The rise can also be found by subtracting the $y\text{-coordinates}$ of the points.

$$\begin{array}{c}{y}_{2}-{y}_{1}\\ 6-3\\ 3\end{array}$$

We counted a run of $5.$ The run can also be found by subtracting the $x\text{-coordinates}.$

$$\begin{array}{c}{x}_{2}-{x}_{1}\\ 7-2\\ 5\end{array}$$

We know | $m=\frac{\text{rise}}{\text{run}}$ |

So | $m=\frac{3}{5}$ |

We rewrite the rise and run by putting in the coordinates. | $m=\frac{6-3}{7-2}$ |

But 6 is the $y$-coordinate of the second point, ${y}_{2}$ and 3 is the $y$-coordinate of the first point ${y}_{1}$. So we can rewrite the rise using subscript notation. | $m=\frac{{y}_{2}-{y}_{1}}{7-2}$ |

Also 7 is the $x$-coordinate of the second point, ${x}_{2}$ and 2 is the $x$-coordinate of the first point ${x}_{2}$. So we rewrite the run using subscript notation. | $m=\frac{{y}_{2}-{y}_{1}}{{x}_{2}-{x}_{1}}$ |

We’ve shown that $m=\frac{{y}_{2}-{y}_{1}}{{x}_{2}-{x}_{1}}$ is really another version of $m=\frac{\text{rise}}{\text{run}}.$ We can use this formula to find the slope of a line when we have two points on the line.

### Slope Formula

The slope of the line between two points $({x}_{1},{y}_{1})$ and $({x}_{2},{y}_{2})$ is

$$m=\frac{{y}_{2}-{y}_{1}}{{x}_{2}-{x}_{1}}$$

Say the formula to yourself to help you remember it:

$$\text{Slope is}\phantom{\rule{0ex}{0ex}}y\phantom{\rule{0ex}{0ex}}\text{of the second point minus}\phantom{\rule{0ex}{0ex}}y\phantom{\rule{0ex}{0ex}}\text{of the first point}$$

$$\text{over}$$

$$x\phantom{\rule{0ex}{0ex}}\text{of the second point minus}\phantom{\rule{0ex}{0ex}}x\phantom{\rule{0ex}{0ex}}\text{of the first point.}$$

### Manipulative Mathematics

### Example 11.38

Find the slope of the line between the points $\left(1,2\right)$ and $\left(4,5\right).$

**Answer**We’ll call $(1,2)$ point #1 and $(4,5)$point #2. $\stackrel{{x}_{1},{y}_{1}}{\left(1,2\right)}\phantom{\rule{0ex}{0ex}}\text{and}\phantom{\rule{0ex}{0ex}}\stackrel{{x}_{2},{y}_{2}}{\left(4,5\right)}$ Use the slope formula. $m=\frac{{y}_{2}-{y}_{1}}{{x}_{2}-{x}_{1}}$ Substitute the values in the slope formula: $y\phantom{\rule{0ex}{0ex}}$ of the second point minus $\phantom{\rule{0ex}{0ex}}y\phantom{\rule{0ex}{0ex}}$ of the first point $m=\frac{5-2}{{x}_{2}-{x}_{1}}$ $x\phantom{\rule{0ex}{0ex}}$ of the second point minus $\phantom{\rule{0ex}{0ex}}x\phantom{\rule{0ex}{0ex}}$ of the first point $m=\frac{5-2}{4-1}$ Simplify the numerator and the denominator. $m=\frac{3}{3}$ $m=1$ Let’s confirm this by counting out the slope on the graph.

The rise is $3$ and the run is $3,$ so

$\begin{array}{}\\ m=\frac{\text{rise}}{\text{run}}\hfill \\ m=\frac{3}{3}\hfill \\ m=1\hfill \end{array}$

### Try It 11.74

Find the slope of the line through the given points: $\left(8,5\right)$ and $\left(6,3\right).$

### Try It 11.75

Find the slope of the line through the given points: $\left(1,5\right)$ and $\left(5,9\right)\text{.}$

How do we know which point to call #1 and which to call #2? Let’s find the slope again, this time switching the names of the points to see what happens. Since we will now be counting the run from right to left, it will be negative.

We’ll call $(4,5)$ point #1 and $(1,2)$ point #2. | $\stackrel{{x}_{1},{y}_{1}}{\left(4,5\right)}\phantom{\rule{0ex}{0ex}}\text{and}\phantom{\rule{0ex}{0ex}}\stackrel{{x}_{2},{y}_{2}}{\left(1,2\right)}$ |

Use the slope formula. | $m=\frac{{y}_{2}-{y}_{1}}{{x}_{2}-{x}_{1}}$ |

Substitute the values in the slope formula: | |

$y\phantom{\rule{0ex}{0ex}}$ of the second point minus $\phantom{\rule{0ex}{0ex}}y\phantom{\rule{0ex}{0ex}}$ of the first point | $m=\frac{2-5}{{x}_{2}-{x}_{1}}$ |

$x\phantom{\rule{0ex}{0ex}}$ of the second point minus $\phantom{\rule{0ex}{0ex}}x\phantom{\rule{0ex}{0ex}}$ of the first point | $m=\frac{2-5}{1-4}$ |

Simplify the numerator and the denominator. | $m=\frac{\mathrm{-3}}{\mathrm{-3}}$ |

$m=1$ |

The slope is the same no matter which order we use the points.

### Example 11.39

Find the slope of the line through the points $\left(\mathrm{-2},\mathrm{-3}\right)$ and $\left(\mathrm{-7},4\right).$

**Answer**We’ll call $(\mathrm{-2},\mathrm{-3})$ point #1 and $(\mathrm{-7},4)$ point #2. $\stackrel{{x}_{1},{y}_{1}}{\left(\mathrm{-2},\mathrm{-3}\right)}\phantom{\rule{0ex}{0ex}}\text{and}\phantom{\rule{0ex}{0ex}}\stackrel{{x}_{2},{y}_{2}}{\left(\mathrm{-7},4\right)}$ Use the slope formula. $m=\frac{{y}_{2}-{y}_{1}}{{x}_{2}-{x}_{1}}$ Substitute the values $y\phantom{\rule{0ex}{0ex}}$ of the second point minus $\phantom{\rule{0ex}{0ex}}y\phantom{\rule{0ex}{0ex}}$ of the first point $m=\frac{4-\left(\mathrm{-3}\right)}{{x}_{2}-{x}_{1}}$ $x\phantom{\rule{0ex}{0ex}}$ of the second point minus $\phantom{\rule{0ex}{0ex}}x\phantom{\rule{0ex}{0ex}}$ of the first point $m=\frac{4-\left(\mathrm{-3}\right)}{\mathrm{-7}-\left(\mathrm{-2}\right)}$ Simplify. $m=\frac{7}{\mathrm{-5}}$ $m=-\frac{7}{5}$ Let’s confirm this on the graph shown.

$\begin{array}{}\\ \\ \\ \phantom{\rule{0ex}{0ex}}m=\frac{\text{rise}}{\text{run}}\\ m=\frac{\mathrm{-7}}{5}\\ \phantom{\rule{0ex}{0ex}}m=-\frac{7}{5}\end{array}$

### Try It 11.76

Find the slope of the line through the pair of points: $\left(\mathrm{-3},4\right)$ and $\left(2,\mathrm{-1}\right).$

### Try It 11.77

Find the slope of the line through the pair of points: $\left(\mathrm{-2},6\right)$ and $\left(\mathrm{-3},\mathrm{-4}\right).$

### Graph a Line Given a Point and the Slope

In this chapter, we graphed lines by plotting points, by using intercepts, and by recognizing horizontal and vertical lines.

Another method we can use to graph lines is the point-slope method. Sometimes, we will be given one point and the slope of the line, instead of its equation. When this happens, we use the definition of slope to draw the graph of the line.

### Example 11.40

Graph the line passing through the point $\left(1,\mathrm{-1}\right)$ whose slope is $m=\frac{3}{4}.$

**Answer**Plot the given point, $\left(1,\mathrm{-1}\right).$

Use the slope formula $m=\frac{\text{rise}}{\text{run}}$ to identify the rise and the run.

$$\begin{array}{}\\ \\ \phantom{\rule{0ex}{0ex}}m=\frac{3}{4}\hfill \\ \frac{\text{rise}}{\text{run}}=\frac{3}{4}\hfill \\ \\ \\ \phantom{\rule{0ex}{0ex}}\text{rise}=3\hfill \\ \phantom{\rule{0ex}{0ex}}\text{run}=4\hfill \end{array}$$

Starting at the point we plotted, count out the rise and run to mark the second point. We count $3$ units up and $4$ 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 $3$ and to the right $4.$ We should get to another point on the line.

### Try It 11.78

Graph the line passing through the point with the given slope:

$\left(2,\mathrm{-2}\right),m=\frac{4}{3}$

### Try It 11.79

Graph the line passing through the point with the given slope:

$\left(\mathrm{-2},3\right),m=\frac{1}{4}$

### How To

#### Graph a line given a point and a slope.

- Step 1. Plot the given point.
- Step 2. Use the slope formula to identify the rise and the run.
- Step 3. Starting at the given point, count out the rise and run to mark the second point.
- Step 4. Connect the points with a line.

### Example 11.41

Graph the line with $y$-intercept $(0,2)$ and slope $m=-\frac{2}{3}.$

**Answer**Plot the given point, the $y$-intercept $(0,2).$

Use the slope formula $m=\frac{\text{rise}}{\text{run}}$ to identify the rise and the run.

$$\begin{array}{}\\ \\ \phantom{\rule{0ex}{0ex}}m=-\frac{2}{3}\hfill \\ \frac{\text{rise}}{\text{run}}=\frac{\mathrm{-2}}{3}\hfill \\ \\ \\ \phantom{\rule{0ex}{0ex}}\text{rise}=\mathrm{\u20132}\hfill \\ \phantom{\rule{0ex}{0ex}}\text{run}=3\hfill \end{array}$$

Starting at $(0,2),$ count the rise and the run and mark the second point.

Connect the points with a line.

### Try It 11.80

Graph the line with the given intercept and slope:

$y$-intercept $4,m=-\frac{5}{2}$

### Try It 11.81

Graph the line with the given intercept and slope:

$x$-intercept $\mathrm{-3},m=-\frac{3}{4}$

### Example 11.42

Graph the line passing through the point $(\mathrm{-1},\mathrm{-3})$ whose slope is $m=4.$

**Answer**Plot the given point.

Identify the rise and the run. $m=4$ Write 4 as a fraction. $\frac{\text{rise}}{\text{run}}=\frac{4}{1}$ $\text{rise}=4\phantom{\rule{0ex}{0ex}}\text{run}=1$ Count the rise and run.

Mark the second point. Connect the two points with a line.

### Try It 11.82

Graph the line passing through the point $\left(\mathrm{-2},1\right)$ and with slope $m=3.$

### Try It 11.83

Graph the line passing through the point $\left(4,\mathrm{-2}\right)$ and with slope $m=\mathrm{-2}.$

### Solve Slope Applications

At the beginning of this section, we said there are many applications of slope in the real world. Let’s look at a few now.

### Example 11.43

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**Use the slope formula. $m=\frac{\text{rise}}{\text{run}}$ Substitute the values for rise and run. $m=\frac{\text{9 ft}}{\text{18 ft}}$ Simplify. $m=\frac{1}{2}$ The slope of the roof is $\frac{1}{2}$.

### Try It 11.84

Find the slope given rise and run: A roof with a rise $=14$ and run $=24.$

### Try It 11.85

Find the slope given rise and run: A roof with a rise $=15$ and run $=36.$

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.

### Example 11.44

Sewage pipes must slope down $\frac{1}{4}$ inch per foot in order to drain properly. What is the required slope?

**Answer**Use the slope formula. $m=\frac{\text{rise}}{\text{run}}$ $m=\frac{-\frac{1}{4}\phantom{\rule{0ex}{0ex}}\text{in}\text{.}}{1\phantom{\rule{0ex}{0ex}}\text{ft}}$ $m=\frac{-\frac{1}{4}\phantom{\rule{0ex}{0ex}}\text{in}\text{.}}{1\phantom{\rule{0ex}{0ex}}\text{ft}}$ Convert 1 foot to 12 inches. $m=\frac{-\frac{1}{4}\phantom{\rule{0ex}{0ex}}\text{in}\text{.}}{12\phantom{\rule{0ex}{0ex}}\text{in.}}$ Simplify. $m=-\frac{1}{48}$ The slope of the pipe is $-\frac{1}{48}.$

### Try It 11.86

Find the slope of the pipe: The pipe slopes down $\frac{1}{3}$ inch per foot.

### Try It 11.87

Find the slope of the pipe: The pipe slopes down $\frac{3}{4}$ inch per yard.

### Media

#### ACCESS ADDITIONAL ONLINE RESOURCES

### Section 11.4 Exercises

#### Practice Makes Perfect

**Use Geoboards to Model Slope**

In the following exercises, find the slope modeled on each geoboard.

203.

204.

205.

206.

In the following exercises, model each slope. Draw a picture to show your results.

207.

$\frac{2}{3}$

208.

$\frac{3}{4}$

209.

$\frac{1}{4}$

210.

$\frac{4}{3}$

211.

$-\frac{1}{2}$

212.

$-\frac{3}{4}$

213.

$-\frac{2}{3}$

214.

$-\frac{3}{2}$

**Find the Slope of a Line from its Graph**

In the following exercises, find the slope of each line shown.

215.

216.

217.

218.

219.

220.

221.

222.

223.

224.

225.

226.

227.

228.

229.

230.

**Find the Slope of Horizontal and Vertical Lines**

In the following exercises, find the slope of each line.

231.

$y=3$

232.

$y=1$

233.

$x=4$

234.

$x=2$

235.

$y=\mathrm{-2}$

236.

$y=\mathrm{-3}$

237.

$x=\mathrm{-5}$

238.

$x=\mathrm{-4}$

**Use the Slope Formula to find the Slope of a Line between Two Points**

In the following exercises, use the slope formula to find the slope of the line between each pair of points.

239.

$\left(1,4\right),\left(3,9\right)$

240.

$\left(2,3\right),\left(5,7\right)$

241.

$\left(0,3\right),\left(4,6\right)$

242.

$\left(0,1\right),\left(5,4\right)$

243.

$\left(2,5\right),\left(4,0\right)$

244.

$\left(3,6\right),\left(8,0\right)$

245.

$\left(\mathrm{-3},3\right),\left(2,\mathrm{-5}\right)$

246.

$\left(\mathrm{-2},4\right),\left(3,\mathrm{-1}\right)$

247.

$\left(\mathrm{-1},\mathrm{-2}\right),\left(2,5\right)$

248.

$\left(\mathrm{-2},\mathrm{-1}\right),\left(6,5\right)$

249.

$\left(4,\mathrm{-5}\right),\left(1,\mathrm{-2}\right)$

250.

$\left(3,\mathrm{-6}\right),\left(2,\mathrm{-2}\right)$

**Graph a Line Given a Point and the Slope**

In the following exercises, graph the line given a point and the slope.

251.

$\left(1,\mathrm{-2}\right);m=\frac{3}{4}$

252.

$\left(1,\mathrm{-1}\right);m=\frac{1}{2}$

253.

$\left(2,5\right);m=-\frac{1}{3}$

254.

$\left(1,4\right);m=-\frac{1}{2}$

255.

$\left(\mathrm{-3},4\right);m=-\frac{3}{2}$

256.

$\left(\mathrm{-2},5\right);m=-\frac{5}{4}$

257.

$\left(\mathrm{-1},\mathrm{-4}\right);m=\frac{4}{3}$

258.

$\left(\mathrm{-3},\mathrm{-5}\right);m=\frac{3}{2}$

259.

$\left(0,3\right);m=-\frac{2}{5}$

260.

$\left(0,5\right);m=-\frac{4}{3}$

261.

$\left(\mathrm{-2},0\right);m=\frac{3}{4}$

262.

$\left(\mathrm{-1},0\right);m=\frac{1}{5}$

263.

$\left(\mathrm{-3},3\right);m=2$

264.

$\left(\mathrm{-4},2\right);m=4$

265.

$\left(1,5\right);m=\mathrm{-3}$

266.

$\left(2,3\right);m=\mathrm{-1}$

**Solve Slope Applications**

In the following exercises, solve these slope applications.

267.

**Slope of a roof** A fairly easy way to determine the slope is to take a $\text{12-inch}$ level and set it on one end on the roof surface. Then take a tape measure or ruler, and measure from the other end of the level down to the roof surface. You can use these measurements to calculate the slope of the roof. What is the slope of the roof in this picture?

268.

What is the slope of the roof shown?

269.

**Road grade** A local road has a grade of $\text{6\%}.$ The grade of a road is its slope expressed as a percent.

- ⓐ Find the slope of the road as a fraction and then simplify the fraction.
- ⓑ What rise and run would reflect this slope or grade?

270.

**Highway grade** A local road rises $2$ feet for every $50$ feet of highway.

- ⓐ What is the slope of the highway?
- ⓑ The grade of a highway is its slope expressed as a percent. What is the grade of this highway?

#### Everyday Math

271.

**Wheelchair ramp** The rules for wheelchair ramps require a maximum $1$ inch rise for a $12$ inch run.

- ⓐ What run must the ramp have to accommodate a $\text{24-inch}$ rise to the door?
- ⓑ Draw a model of this ramp.

272.

**Wheelchair ramp** A $\text{1-inch}$ rise for a $\text{16-inch}$ run makes it easier for the wheelchair rider to ascend the ramp.

- ⓐ What run must the ramp have to easily accommodate a $\text{24-inch}$ rise to the door?
- ⓑ Draw a model of this ramp.

#### Writing Exercises

273.

What does the sign of the slope tell you about a line?

274.

How does the graph of a line with slope $m=\frac{1}{2}$ differ from the graph of a line with slope $m=2?$

275.

Why is the slope of a vertical line undefined?

276.

Explain how you can graph a line given a point and its slope.

#### Self Check

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

ⓑ On a scale of 1–10, how would you rate your mastery of this section in light of your responses on the checklist? How can you improve this?

## FAQs

### How do you read the slope of a line? ›

**Using the Slope Equation**

- Pick two points on the line and determine their coordinates.
- Determine the difference in y-coordinates of these two points (rise).
- Determine the difference in x-coordinates for these two points (run).
- Divide the difference in y-coordinates by the difference in x-coordinates (rise/run or slope).

**How to calculate a slope? ›**

Percent of slope is determined by **dividing the amount of elevation change by the amount of horizontal distance covered (sometimes referred to as "the rise divided by the run"), and then multiplying the result by 100**.

**What is the slope of a line answers? ›**

The slope equals the rise divided by the run: **Slope =riserun** Slope = rise run . You can determine the slope of a line from its graph by looking at the rise and run. One characteristic of a line is that its slope is constant all the way along it.

**How do you write an answer for slope? ›**

One can use either the slope formula **m = (y _{2} – y_{1})/(x_{2} – x_{1})** or the standard line equation, y = mx + b to solve for the slope, m.

**How do you read slope and intercept? ›**

The equation of the line is written in the slope-intercept form, which is: y = mx + b, where m represents the slope and b represents the y-intercept. In our equation, y = 6x + 2, we see that the slope of the line is 6.

**What is an example of a slope? ›**

The m is the slope of the line. And b is the b in the point that is the y-intercept (0, b). For example, for the equation **y = 3x – 7**, the slope is 3, and the y-intercept is (0, −7).

**What are the 3 formulas for slope? ›**

Fact | You can use this fact when you know: |
---|---|

Slope formula: | two points on a line |

Slope-intercept formula: y = mx + b | the slope and y-intercept of a line |

Point-slope formula: y - y_{1} = m(x - x_{1}) | the slope of a line and a point on the line |

Parallel lines have equal slopes | the slope of a line |

**What does a slope represent? ›**

Slope measures **the rate of change in the dependent variable as the independent variable changes**. Mathematicians and economists often use the Greek capital letter D or D as the symbol for change. Slope shows the change in y or the change on the vertical axis versus the change in x or the change on the horizontal axis.

**What is line slope grade? ›**

The grade (also called slope, incline, gradient, mainfall, pitch or rise) of a physical feature, landform or constructed line refers to **the tangent of the angle of that surface to the horizontal**. It is a special case of the slope, where zero indicates horizontality.

**How do you find the equation of a line? ›**

**How to Find the Equation of a Line from Two Points**

- Find the slope using the slope formula. ...
- Use the slope and one of the points to solve for the y-intercept (b). ...
- Once you know the value for m and the value for b, you can plug these into the slope-intercept form of a line (y = mx + b) to get the equation for the line.

### How do you find the slope of two points? ›

Use the slope formula to find the slope of a line given the coordinates of two points on the line. The slope formula is **m=(y2-y1)/(x2-x1)**, or the change in the y values over the change in the x values. The coordinates of the first point represent x1 and y1.

**How do you explain slope to students? ›**

**Slope is calculated as the ratio of the amount of vertical change to horizontal change**. The easiest way to remember this is that slope is equal to "Rise over Run." For example, if the slope of a line is 2/3, that means the line goes up 2 units for every 3 units to the right. Up 2, over 3.

**What is slope in simple? ›**

1. : **ground that forms a natural or artificial incline**. : upward or downward slant or inclination or degree of slant.

**What are the 4 types of slope? ›**

Slopes come in 4 different types: **negative, positive, zero, and undefined**. as x increases. The slope of a line can also be interpreted as the “average rate of change”. It tells us how fast y is changing with respect to x.

**How do you do slope step by step? ›**

**There are three steps in calculating the slope of a straight line when you are not given its equation.**

- Step One: Identify two points on the line.
- Step Two: Select one to be (x1, y1) and the other to be (x2, y2).
- Step Three: Use the slope equation to calculate slope.

**Is the slope the same as y-intercept? ›**

Slope and y-Intercept Values

**The slope indicates the rate of change in y per unit change in x**. The y-intercept indicates the y-value when the x-value is 0.

**How do you find the y-intercept? ›**

Since the y-intercept always has a corresponding x-value of 0, replace x with 0 in the equation and solve for y. On a graph, the y-intercept can be found by **finding the value of y when x=0**. This is the point at which the graph crosses through the y-axis.

**What is slope in math facts? ›**

The slope is defined as **the ratio of the vertical change between two points, the rise, to the horizontal change between the same two points, the run**. The slope of a line is usually represented by the letter m. (x_{1}, y_{1}) represents the first point whereas (x_{2}, y_{2}) represents the second point.

**Can a slope be negative? ›**

**A slope can be negative when the two variables are inversely related to one another**, as one increases, the other decreases. You can find the slope of a line by looking at a graph, looking at an equation, or using the slope formula.

**Why is it called slope? ›**

Language experts believe the word slope **came from the Middle English word aslope, an adverb that means “at an angle.”** The word has a noun form you can use for something that is at an angle — on a slope — like a steep hill or the ramp in a parking garage.

### How steep is a 15 degree slope? ›

DEGREES | GRADIENT | PERCENT |
---|---|---|

5° | 1 : 11.43 | 8.75% |

10° | 1 : 5.67 | 17.63% |

15° | 1 : 3.73 | 26.79% |

30° | 1 : 1.73 | 57.74% |

**What does a 1 in 20 slope mean? ›**

Minimum slope: 1:20, meaning you will have **20 units of horizontal length (run) by 1 unit of vertical length (rise)**.

**What is a 100% slope? ›**

**A rise of 100 feet over a run of 100 feet** yields a 100 percent slope. A 50-foot rise over a 100-foot run yields a 50 percent slope. Another way to express slope is as a slope angle, or degree of slope.

**What is the formula for the slope of a graph? ›**

The slope from graph can be calculated by picking any two points on it and applying the formula rise/run. It can be also found by picking two points and applying the formula **(y₂ - y₁) / (x₂ - x₁)**.

**What is the formula for slope distance? ›**

The slope length is calculated using the Pythagorean Theorem, where the vertical distance is the rise and the horizontal distance is the run: **rise ^{2} + run^{2} = slope length^{2}**.

**What is slope in math? ›**

The slope of a line is **a measure of its steepness**. Mathematically, slope is calculated as "rise over run" (change in y divided by change in x).

**How do you calculate ramp slope? ›**

**Slope expressed as a percentage = (h/d) x 100**

An existing ramp of 1 meter in height with a horizontal distance of 10 meters, will have a slope of 10%. Save this picture! Accordingly, knowing the height to overcome on a project allows us to review recommendations for a slope consistent with the required needs.