### learning successes

- Find the slope from a chart
- Identify Rise and Run from a chart
- Distinguishing between graphs of negative and positive slope lines

- Find the slope of two points
- Use the slope formula to define the slope of a line through two points

- Find the slope of the horizontal and vertical lines
- Find the slope of the lines [latex]x=a[/latex] and [latex]y=b[/latex]
- Notice that horizontal lines have slope = 0
- Note that vertical lines have slopes that are undefined

- Identify slopes of parallel and perpendicular lines
- Given a line, identify the slope of another line parallel to it.
- Given a line, identify the slope of another line perpendicular to it.

- Interpret slope in equations and graphs
- Check the slope of a linear equation given a data set
- Interpret the slope of a linear equation as it applies to a real-world situation

## Identify the slope from a chart

The mathematical definition of**Earring**it is very similar to our everyday. In mathematics, slope is used to describe the steepness and direction of lines. By simply looking at the graph of a line, you can learn quite a bit about its slope, particularly in relation to other lines graphed on the same coordinate plane. Consider the graphs of the three lines shown below:

Let's look at lines A and B first. If you picture these lines as hills, you would say that line B is steeper than line A. Line B has a steeper slope than line A.

Then notice that lines A and B rise as you move from left to right. We say that these two lines have a positive slope. Line C slopes down from left to right. Line C has a negative slope. Using two of the points on the line, you can find the slope of the line by finding the slope and the slope. The vertical change between two points is called**lift**, and the horizontal change is called**run**. The slope is equal to the slope divided by the slope: [latex] \displaystyle \text{Slope }=\frac{\text{rise}}{\text{run}}[/latex].

You can determine the slope of a line from its graph by looking at the rise and the slope. A property of a line is that its slope is constant throughout its length. So you can pick any 2 points along the graph of the line to find the slope. Let's look at an example.

### Example

Use the chart to find the slope of the line.

show solution

This line has a slope of [latex] \displaystyle \frac{1}{2}[/latex] no matter which two points you pick on the line. Try to measure the slope from the origin [latex](0,0)[/latex] to the point [latex](6,3)[/latex]. You will find that [latex]\text{rise}=3[/latex] and [latex]\text{run}=6[/latex]. The slope is [latex] \displaystyle \frac{\text{rise}}{\text{run}}=\frac{3}{6}=\frac{1}{2}[/latex]. Is the same!

Let's look at another example.

### Example

Use the chart to find the slope of the two lines.

show solution

If you look at the two lines, you can see that the blue line is steeper than the red line. It makes sense that the value of the blue line's slope, 4, would be greater than the value of the red line's slope, [latex] \displaystyle \frac{1}{4}[/latex]. The greater the slope, the steeper the line becomes.

## Finding the slope of a straight line from a graph

## Distinguishing between graphs of negative and positive slope lines

Direction is important when trying to determine grade. It's important to pay attention to whether you're moving up, down, left, or right; that is, whether you are moving in a positive or negative direction. If you raise to reach your second point, the raise is positive. If you go down to get to your second point, the rise is negative. If you go right to get to your second point, the run is positive. If you go left to get to the second point, the run is negative.

In the next two examples you can see a positive slope and a negative slope.

### Example (Advanced)

Find the slope of the line graphed below.

show solution

The example below shows a line with a negative slope.

### Example

Find the slope of the line graphed below.

show solution

In the example above, you could have found the slope by starting at point B, doing [latex]{-2}[/latex], and then [latex]+3[/latex] going up to get to point A The result is still a slope of [latex]\displaystyle\frac{\text{rise}}{\text{run}}=\frac{+3}{-2}=-\frac{3}{2}[ /Latex].

## Find the slope of two points on the line

You've seen that you can find the slope of a line on a chart by measuring the slope and the slope. You can also find the slope of a line without its graph if you know the coordinates of any two points on that line. Each point has a set of coordinates: a*X*-value and a*j*-value written as an ordered pair (*X*,*j*). Is*X*The value indicates where a point is located horizontally. He*j*The value indicates where the point is located vertically.

Imagine two points on a line: point 1 and point 2. Point 1 has coordinates [latex]\left(x_{1},y_{1}\right)[/latex] and point 2 has coordinates [ latex]\ left( x_{2},y_{2}\right)[/latex].

The height is the vertical distance between the two points, that is the difference between them*j*-Coordinates. This makes the slope [latex]\left(y_{2}-y_{1}\right)[/latex]. The run between these two points is the difference in the*X*-coordenadas, o [latex]\left(x_{2}-x_{1}\right)[/latex].

Entons, [latex] \displaystyle \text{Slope}=\frac{\text{rise}}{\text{run}}[/latex] o [latex] \displaystyle m=\frac{{{y}_{ 2}}-{{y}_{1}}}{{{x}_{2}}-{{x}_{1}}}[/latex]

In the example below, you can see that the line has two points, each shown as an ordered pair. Point [latex](0,2)[/latex] is displayed as point 1 and [latex](−2,6)[/latex] is displayed as point 2. So you go from point 1 to point 2. A triangle is drawn above the line to illustrate the climb and run.

You can see on the chart that the height from point 1 to point 2 is 4 because you are moving 4 units in a positive (up) direction. The run is [latex]−2[/latex], because then you move 2 units in the negative direction (left). With the slope formula

[latex] \displaystyle \text{Pendiente}=\frac{\text{rise}}{\text{run}}=\frac{4}{-2}=-2[/latex].

You don't need the graph to find the slope. You can just use the coordinates and track exactly which point is 1 and which is point 2. Let's organize the information related to the two points:

Name | orderly pair | coordinates |
---|---|---|

point 1 | [Latex](0,2)[/Latex] | [latex]\begin{matrix}{l}x_{1}=0\\y_{1}=2\end{matrix}[/latex] |

point 2 | [Latex](−2,6)[/Latex] | [latex]\begin{matrix}{l}x_{2}=-2\\y_{2}=6\end{matrix}[/latex] |

The slope, [latex]m=\frac{y_{2}-y_{1}}{x_{2}-x_{1}}=\frac{6-2}{-2-0}=\frac{ 4}{-2}=-2[/latex]. the slope of the line,*Metro*, ist [latex]−2[/latex].

It doesn't matter which point is called point 1 and which one is called point 2. You could also have named [latex](−2,6)[/latex] point 1 and [latex](0,2)[/latex ] point 2 In this case, substituting the coordinates into the slope formula gives the equation [latex ]m=\frac{2-6}{0-\left(-2\right)}=\frac{-4}{2 }=-2[/latex]. Here, too, the slope is [latex]m=-2[/latex]. This is the same slope as before. The most important thing is to be consistent when subtracting: you should always subtract in the same order [latex]\left(y_{2},y_{1}\right)[/latex]_{}y [latex]\left(x_{2},x_{1}\right)[/latex].

### Example

What is the slope of the line containing the points [latex](5,5)[/latex] and [latex](4,2)[/latex]?

show solution

The example below shows the solution if you reverse the order of the points and call [latex](5,5)[/latex] point 1 and [latex](4,2)[/latex] point 2.

### Example

What is the slope of the line containing the points [latex](5,5)[/latex] and [latex](4,2)[/latex]?

show solution

Note that no matter which ordered pair is called point 1 and which is called point 2, the slope is still 3.

### Example (Advanced)

What is the slope of the line containing the points [latex](3,-6.25)[/latex] and [latex](-1,8.5)[/latex]?

show solution

Imagine a horizontal line on a chart. No matter which two points you choose on the line, they will always be the same*j*-coordinate. The equation for this line is [latex]y=3[/latex]. The equation can also be written as [latex]y=\left(0\right)x+3[/latex].

## Video: Finding the slope of a line at two points on the line

## Finding the slopes of horizontal and vertical lines

So far you have considered lines that go "uphill" or "downhill". Their slopes can be steep or flat, but they are always positive or negative numbers. But there are two other types of lines, horizontal and vertical. What is the slope of a flat line or level ground? On a wall or on a vertical line?

Using the form [latex]y=0x+3[/latex] you can see that the slope is 0. You can also use the slope formula with two points on that horizontal line to find the slope of that horizontal line. Using [latex](−3,3)[/latex] as point 1 and (2, 3) as point 2, you get:

[latex] \displaystyle \begin{array}{l}m=\frac{{{y}_{2}}-{{y}_{1}}}{{{x}_{2}}-{ {x}_{1}}}\\\\m=\frac{3-3}{2-\left(-3\right)}=\frac{0}{5}=0\end{matriz} [/Latex]

The slope of this horizontal line is 0.

Consider a horizontal line. No matter which two points you choose on the line, they will always be the same*j*-coordinate. So when you apply the slope formula, the numerator is always 0. Zero divided by any non-zero number is 0, so the slope of any horizontal line is always 0.

The equation for the horizontal line [latex]y=3[/latex] tells you that whatever two points you choose on that line, the*y-*The coordinate is always 3.

What about the vertical lines? In your case, no matter which two points you choose, they will always be the same*X*-coordinate. The equation for this line is [latex]x=2[/latex].

There is no way to put this equation in the form of a slope point, such as B. the coefficient of*j*es [latex]0\left(x=0y+2\right)[/latex].

So what happens when you use the slope formula with two points on that vertical line to find the slope? Using [latex](2,1)[/latex] as point 1 and [latex](2,3)[/latex] as point 2 you get:

[latex] \displaystyle \begin{array}{l}m=\frac{{{y}_{2}}-{{y}_{1}}}{{{x}_{2}}-{ {x}_{1}}}\\\\m=\frac{3-1}{2-2}=\frac{2}{0}\end{matrix}[/latex]

But dividing by zero makes no sense for the set of real numbers. Because of this fact, the slope of this vertical line is said to be undefined. This applies to all vertical lines: they all have an undefined slope.

### Example

What is the slope of the line containing the points [latex](3,2)[/latex] and [latex](−8,2)[/latex]?

show solution

## Find slopes of horizontal and vertical lines

## Characterize the slopes of parallel and perpendicular lines

When you graph two or more linear equations on a coordinate plane, they usually intersect at a point. However, if two lines never intersect in a coordinate plane, they will be called**parallel lines**. You will also see the case where two lines intersect at right angles in a coordinate plane. These are called**vertical lines**. In each of these cases, the gradients of the graphs have a special relationship to one another.

Parallel lines are two or more lines in a plane that never intersect. Examples of parallel lines surround us, like the opposite sides of a rectangular frame and the shelves of a bookshelf.

Perpendicular lines are two or more lines that intersect at a 90-degree angle, like the two lines drawn in this diagram. These 90 degree angles are also known as right angles.

Vertical lines are everywhere too, not just on graph paper but in the world around us, from the crossing pattern at an intersection to the colored lines on a plaid shirt.

### parallel lines

Two non-perpendicular lines in a plane are parallel if both have:

- the same slope
- anders
*j*-Crossroads

Any two perpendicular lines in a plane are parallel.

### Example

Find the slope of a line parallel to the line [latex]y=−3x+4[/latex].

show solution

### Example

Determines whether the lines [latex]y=6x+5[/latex] and [latex]y=6x–1[/latex] are parallel.

show solution

### vertical lines

Two non-vertical lines are perpendicular if the slope of one is the negative reciprocal of the slope of the other. If the slope of the first equation is 4, then the slope of the second equation must be [latex]-\frac{1}{4}[/latex] for the lines to be perpendicular.

You can also check the two slopes to see if the lines are perpendicular by multiplying the two slopes. If they are perpendicular, the product of the slopes is [latex]−1[/latex]. Example: [latex] 4\cdot-\frac{1}{4}=\frac{4}{1}\cdot-\frac{1}{4}=-1[/latex].

### Example

Find the slope of a line perpendicular to the line [latex]y=2x–6[/latex].

show solution

To find the slope of a vertical line, find the reciprocal [latex] \displaystyle \tfrac{1}{2}[/latex] and then the inverse of that reciprocal [latex] \displaystyle -\tfrac{1} {2}[ /latex].

Note that the product [latex]2\left(-\frac{1}{2}\right)=\frac{2}{1}\left(-\frac{1}{2}\right)= -1 [/latex], so that means the slopes are vertical.

In case one of the lines is vertical, the slope of that line is undefined and it is not possible to calculate the product with an undefined number. If a line is vertical, the line perpendicular to it is horizontal and has no slope ([latex]m=0[/latex]).

### Example

Determine if the lines [latex]y=−8x+5[/latex]y [latex] \displaystyle y\,\text{=}\,\,\frac{1}{8}x-1[/latex ] are parallel, perpendicular, or neither.

show solution

## The slope of parallel and perpendicular lines

## Check the slope from a data set

Various institutions and groups collect huge amounts of data every day. This data is used for many purposes, including business decisions about location and marketing, government decisions about resource allocation and infrastructure, and personal decisions about where to live or where to buy food.

The example below shows how a data set can be used to define the slope of a linear equation.

### Example

Using the data set, check the values of the slopes of each equation.

Linear equations describing the change in median house values between 1950 and 2000 in Mississippi and Hawaii are as follows:

**Hawaii:**[latex]y=3966x+74.400[/latex]

**Mississippi:**[latex]y=924x+25.200[/latex]

The equations are based on the following dataset.

x = the number of years since 1950; and y = the median home value in each state.

Year (X) | Hauswert in Mississippi (j) | Hauswert in Hawaii (j) |
---|---|---|

0 | 25.200 $ | 74.400 $ |

50 | 71.400 $ | 272.700 $ |

The slopes of each equation can be calculated using the formula you learned in the Slopes section.

[latex] \displaystyle m=\frac{{{y}_{2}}-{{y}_{1}}}{{{x}_{2}}-{{x}_{1}} }[/Latex]

**Mississippi:**

Name | orderly pair | coordinates |
---|---|---|

point 1 | (0, 25.200) | [latex]\begin{matrix}{l}x_{1}=0\\y_{1}=25.200\end{matrix}[/latex] |

point 2 | (50, 71.400) | [latex]\begin{matrix}{l}x_{2}=50\\y_{2}=71.400\end{matrix}[/latex] |

[latex] \displaystyle m=\frac{{71 400}-{25 200}}{{50}-{0}}=\frac{{46 200}}{{50}} = 924[/latex]

We verified that the slope [latex] \displaystyle m = 924[/latex] matches the provided dataset.

**Hawaii:**

Name | orderly pair | coordinates |
---|---|---|

point 1 | (0, 74.400) | [latex]\begin{matrix}{l}x_{1}=1950\\y_{1}=74 400\end{matrix}[/latex] |

point 2 | (50, 272,700) | [latex]\begin{matrix}{l}x_{2}=2000\\y_{2}=272.700\end{matrix}[/latex] |

[latex]\displaystyle m=\frac{{272 700}-{74 400}}{{50}-{0}}=\frac{{198 300}}{{50}} = 3966[/latex]

We verified that the slope [latex] \displaystyle m = 3966[/latex] matches the provided data set.

### Example

Using the data set, check the values of the slopes in the equation.

A linear equation describing the change in the number of high school students smoking in a group of 100 between 2011 and 2015 is as follows:

[latex]y = -1,75x+16[/latex]

And it's based on the data in this table provided by the Centers for Disease Control.

x = the number of years since 2011; and y = the number of high school smokers per 100 students.

Year | Number of high school students who smoke cigarettes (per 100) |

0 | sixteen |

4 | 9 |

Name | orderly pair | coordinates |
---|---|---|

point 1 | (0, 16) | [latex]\begin{matrix}{l}x_{1}=0\\y_{1}=16\end{matrix}[/latex] |

point 2 | (4, 9) | [latex]\begin{matrix}{l}x_{2}=4\\y_{2}=9\end{matrix}[/latex] |

[latex] \displaystyle m=\frac{{{y}_{2}}-{{y}_{1}}}{{{x}_{2}}-{{x}_{1}} }=\frac{{9-16}}{{4-0}} =\frac{{-7}}{{4}}=-1,75[/látex]

We verified that the slope [latex] \displaystyle{m=-1.75}[/latex] matches the provided dataset.

## Interpret the slope of the linear equation

OK, now we've verified that the data can give us the slope of a linear equation. So what? We can use this information to describe in words how something is changing.

First, let's look at the different types of slopes that are possible in a linear equation.

We often use specific words to describe the different types of slopes when using lines and equations to represent "real world" situations. The table below compares the nature of the slope to the common language used to describe it both verbally and visually.

incline type | visual description | verbal description |

positive | uphill | growing |

Negative | downhill | decreasing |

0 | horizontal | Constantly |

Not defined | Vertical | N / A |

### Example

Explain in words the slope of each equation for house values.

**Hawaii:**[latex]y = 3966x+74.400[/latex]

**Mississippi:**[latex]y = 924x+25.200[/latex]

show solution

## Interpreting the meaning of the slope in a linear equation: home averages

### Example

Interpret in words the slope of the line describing the change in the number of high school smokers.

Apply units to the slope formula. He*X*Values represent years, and the*j*Values represent the number of smokers. Keep in mind that this dataset is per 100 high school students.

[latex] \displaystyle m=\frac{{9-16}}{{2015-2011}} =\frac{{-7 \text{ smoker}}}{{4\text{ year}}}=-1 .75 \frac{\text{ smoker}}{\text{ year}}[/latex]

The slope of this linear equation is**Negative**, then this tells us that there is a**reduce**in the number of school-age smokers each year.

The number of high school students who smoke is falling by 1.75 per 100 each year.

## Interpret the meaning of the slope of a linear equation: smokers

On the next page we will see how to interpret them*j*-Intersection of a linear equation, and make a prediction based on a linear equation.

## Summary

Slope describes the steepness of a line. The slope of each line remains constant along the line. The slope can also give you information about the direction of the line in the coordinate plane. The slope can be calculated by looking at the graph of a line or by using the coordinates of any two points on a line. There are two common formulas for slope: [latex] \displaystyle \text{Slope }=\frac{\text{rise}}{\text{run}}[/latex] and [latex] \displaystyle m=\frac { {{y}_{2}}-{{y}_{1}}}{{{x}_{2}}-{{x}_{1}}}[/latex] where [latex] m =\text{slope}[/latex]y [latex] \displaystyle ({{x}_{1}},{{y}_{1}})[/latex] y [latex] \displaystyle ({ { x}_{2}},{{y}_{2}})[/latex] are two points on the line.

The following images summarize the slopes of different types of lines.