Lesson Objectives

- Demonstrate an understanding of how to graph a linear equation in two variables
- Learn how to find the slope of a line using the slope formula
- Learn how to find the slope of a line from its graph
- Learn about special case scenarios: the slope of a horizontal and vertical line
- Learn how to find the slope of a line from its equation
- Learn how to determine if a line is positively sloped or negatively sloped

## How to Find the Slope of a Line

In our last lesson, we learned how to graph a linear equation in two variables. The concept of slope is very important in Algebra and higher-level math. The slope of a line is a measure of the line's steepness. Officially, the slope of a line is the ratio of the vertical change, known as the "rise" to the horizontal change, known as the "run" as we move along the line from one point to another. We normally use the lowercase letter "m" to denote slope: $$m=\frac{\text{rise}}{\text{run}}$$ When we think about the rise or change in y-values, this can be positive, with a move up, or negative, with a move down. Similarly, with the run or change in x-values, this can be positive, with a move to the right, or negative, with a move left. If we want to get more specific, the slope of a line is the change in y per one-unit change in x. If we had a line whose slope is 7, this translates into: $$m=\frac{\text{rise}}{\text{run}}=\frac{7}{1}$$ For this line, the line rises 7 units for each 1 unit move to the right.

As another example, suppose we had a line with a slope of 3/2. What does this translate into? $$m=\frac{\text{rise}}{\text{run}}=\frac{3}{2}$$ This means our line will rise 3 units for every 2 units that we move to the right. If we want to know how much y will change per a one-unit change in x, we need to obtain a denominator of 1. We can always do this by dividing the numerator by the denominator. $$m=\frac{\text{rise}}{\text{run}}=\frac{3}{2}=1.5$$ This tells us that our line rises 1.5 units for each 1 unit move to the right.

Example 1: Use the slope formula to obtain the slope of the line.

The line that passes through the points: (0,-6) and (3,0).

First, we pick one point and label it as: (x

x

y

x

y

Now let's plug into our formula: $$m=\frac{y_{2}- y_{1}}{x_{2}- x_{1}}$$ $$m=\frac{0 - (-6)}{3 - 0}$$ $$m=\frac{0 + 6}{3}$$ $$m=\frac{6}{3}$$ $$m=2$$ A slope of 2 tells us the line will rise 2 units for every one unit move to the right.

A quick note, many students ask if it matters which point is labeled as point 1 and which is labeled as point 2. It will make no difference with regard to the final calculation. We could have labeled (3,0) as our first point and (0,-6) as the second. The slope will be 2 either way. $$m=\frac{-6 - 0}{0 - 3}$$ $$m=\frac{-6}{-3}$$ $$m=2$$ Example 2: Use the slope formula to obtain the slope of the line.

The line that passes through the points: (0,-3) and (7,-2).

First, we pick one point and label it as: (x

x

y

x

y

Now let's plug into our formula: $$m=\frac{y_{2}- y_{1}}{x_{2}- x_{1}}$$ $$m=\frac{-2 - (-3)}{7 - 0}$$ $$m=\frac{-2 + 3}{7}$$ $$m=\frac{1}{7}$$ A slope of 1/7 tells us the line will rise 1 unit for every 7 units that we move to the right.

Example 3: Find the slope of the line from its graph.

-2x + y = 4 Let's pick two points on the line (-2,0) and (0,4). What is the rise or vertical change from one point to the other? What is the run or horizontal change from one point to the other? We can see that moving from the point (-2,0) to the point (0,4) results in a rise of 4 units and a run of 2 units. This means our slope will be 2. We can see the same result if we plug the points into the slope formula. Let's label (-2,0) as point 1, and (0,4) as point 2: $$m=\frac{4 - 0}{0 - (-2)}$$ $$m=\frac{4}{2}$$ $$m=2$$

Example 4: Use the slope formula to obtain the slope of the line.

The line that passes through the points: (7, -2) and (-7, -2).

We can tell this is a horizontal line since the two y-values are the same (-2). Let's label (7,-2) as point 1 and (-7,-2) as point 2. $$m=\frac{-2 - (-2)}{-7 - 7}$$ $$m=\frac{-2 + 2}{-14}$$ $$m=\frac{0}{-14}$$ $$m=0$$ We can see that our slope is 0. There is no rise when moving along the line from the point (7,-2) to the point (-7,-2) since this is a horizontal line.

Example 5: Use the slope formula to obtain the slope of the line.

The line that passes through the points: (-4,6) and (-4, -6)

We can tell this is a vertical line since the two x-values are the same (-4). Let's label (-4,6) as point 1 and (-4,-6) as point 2. $$m=\frac{-6 - 6}{-4 - (-4)}$$ $$m=\frac{-12}{-4 + 4}$$ $$m=\frac{-12}{0}$$ m, the slope is undefined since we can't divide by 0.

When an equation is in slope-intercept form, m, the coefficient of x is the slope, and b, the constant term is the y-intercept. Let's look at an example.

Example 6: Find the slope of the line by placing the equation in slope-intercept form.

4x - y = -20

Solve the equation for y:

4x - y = -20

-y = -4x - 20

y = 4x + 20

When our equation is in slope-intercept form, the slope is given to us as the coefficient of the x variable. In this case, our coefficient of x is 4, this means the slope of the line is 4. We can prove this by choosing two points on the line and using our slope formula. We know one point would be the y-intercept of (0,20). As another point, we could plug in a 0 for y and solve for x.

0 = 4x + 20

-4x = 20

x = -5

This gives us an ordered pair of (-5,0). Let's plug these two points into our slope formula. We will label (0,20) as point 1 and (-5,0) as point 2: $$m=\frac{0 - 20}{-5 - 0}=\frac{-20}{-5}=4$$ We can see that our slope is 4 either way. Moving forward, we will place an equation in slope-intercept form to obtain the slope since it is a much faster approach.

As another example, suppose we had a line with a slope of 3/2. What does this translate into? $$m=\frac{\text{rise}}{\text{run}}=\frac{3}{2}$$ This means our line will rise 3 units for every 2 units that we move to the right. If we want to know how much y will change per a one-unit change in x, we need to obtain a denominator of 1. We can always do this by dividing the numerator by the denominator. $$m=\frac{\text{rise}}{\text{run}}=\frac{3}{2}=1.5$$ This tells us that our line rises 1.5 units for each 1 unit move to the right.

### Slope Formula

One method that can be used to find the slope of a line is known as the slope formula. $$m=\frac{y_{2}- y_{1}}{x_{2}- x_{1}}$$ $$x_{2}- x_{1}≠ 0$$ To break down our formula, we start with the lowercase letter "m". We mentioned earlier, the lowercase letter "m" is used to denote slope. The x and y variables are meant to represent data obtained from two points (ordered pairs) on the line. In other words, we can take any points on the line and label one point as: (x_{1}, y_{1}) and the other as: (x_{2}, y_{2}). The 1 and 2 in these ordered pairs are known as subscripts. These are read as ("x-sub-one", "y-sub-one"), ("x-sub-two", "y-sub-two"). We can then plug into the slope formula and obtain the slope of our line. Let's look at an example.Example 1: Use the slope formula to obtain the slope of the line.

The line that passes through the points: (0,-6) and (3,0).

First, we pick one point and label it as: (x

_{1}, y_{1}), then the other point will be labeled as: (x_{2}, y_{2}). Let's use (0,-6) as our first point and (3,0) as our second point.x

_{1}= 0y

_{1}= -6x

_{2}= 3y

_{2}= 0Now let's plug into our formula: $$m=\frac{y_{2}- y_{1}}{x_{2}- x_{1}}$$ $$m=\frac{0 - (-6)}{3 - 0}$$ $$m=\frac{0 + 6}{3}$$ $$m=\frac{6}{3}$$ $$m=2$$ A slope of 2 tells us the line will rise 2 units for every one unit move to the right.

A quick note, many students ask if it matters which point is labeled as point 1 and which is labeled as point 2. It will make no difference with regard to the final calculation. We could have labeled (3,0) as our first point and (0,-6) as the second. The slope will be 2 either way. $$m=\frac{-6 - 0}{0 - 3}$$ $$m=\frac{-6}{-3}$$ $$m=2$$ Example 2: Use the slope formula to obtain the slope of the line.

The line that passes through the points: (0,-3) and (7,-2).

First, we pick one point and label it as: (x

_{1}, y_{1}), then the other point will be labeled as: (x_{2}, y_{2}). Let's use (0,-3) as our first point and (7,-2) as our second point.x

_{1}= 0y

_{1}= -3x

_{2}= 7y

_{2}= -2Now let's plug into our formula: $$m=\frac{y_{2}- y_{1}}{x_{2}- x_{1}}$$ $$m=\frac{-2 - (-3)}{7 - 0}$$ $$m=\frac{-2 + 3}{7}$$ $$m=\frac{1}{7}$$ A slope of 1/7 tells us the line will rise 1 unit for every 7 units that we move to the right.

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

We can also find the slope of a line from its graph. We follow the same logic of rise/run. We can find any two points on the graph and count the number of units we rise and run to get from one point to the other. Let's look at an example.Example 3: Find the slope of the line from its graph.

-2x + y = 4 Let's pick two points on the line (-2,0) and (0,4). What is the rise or vertical change from one point to the other? What is the run or horizontal change from one point to the other? We can see that moving from the point (-2,0) to the point (0,4) results in a rise of 4 units and a run of 2 units. This means our slope will be 2. We can see the same result if we plug the points into the slope formula. Let's label (-2,0) as point 1, and (0,4) as point 2: $$m=\frac{4 - 0}{0 - (-2)}$$ $$m=\frac{4}{2}$$ $$m=2$$

### The Slope of a Horizontal Line

When we have a horizontal line such as: y = k, the slope will be 0. This is because the rise or vertical change is always 0, no matter the run or horizontal change. Let's look at an example.Example 4: Use the slope formula to obtain the slope of the line.

The line that passes through the points: (7, -2) and (-7, -2).

We can tell this is a horizontal line since the two y-values are the same (-2). Let's label (7,-2) as point 1 and (-7,-2) as point 2. $$m=\frac{-2 - (-2)}{-7 - 7}$$ $$m=\frac{-2 + 2}{-14}$$ $$m=\frac{0}{-14}$$ $$m=0$$ We can see that our slope is 0. There is no rise when moving along the line from the point (7,-2) to the point (-7,-2) since this is a horizontal line.

### The Slope of a Vertical Line

When we have a vertical line, the slope is "undefined". This is due to the lack of change in x-values. This means our slope formula will involve division by zero, which is undefined. Let's look at an example.Example 5: Use the slope formula to obtain the slope of the line.

The line that passes through the points: (-4,6) and (-4, -6)

We can tell this is a vertical line since the two x-values are the same (-4). Let's label (-4,6) as point 1 and (-4,-6) as point 2. $$m=\frac{-6 - 6}{-4 - (-4)}$$ $$m=\frac{-12}{-4 + 4}$$ $$m=\frac{-12}{0}$$ m, the slope is undefined since we can't divide by 0.

### How to Find the Slope of a Line from an Equation

Once we understand how to find the slope of a line using the slope formula, we can move into using a quicker method. When we solve a linear equation in two variables for y, we are placing the equation in what is known as slope-intercept form. The slope-intercept form of a line lets us know the slope and y-intercept from a simple inspection of the line.#### Slope-Intercept Form:

y = mx + bWhen an equation is in slope-intercept form, m, the coefficient of x is the slope, and b, the constant term is the y-intercept. Let's look at an example.

Example 6: Find the slope of the line by placing the equation in slope-intercept form.

4x - y = -20

Solve the equation for y:

4x - y = -20

-y = -4x - 20

y = 4x + 20

When our equation is in slope-intercept form, the slope is given to us as the coefficient of the x variable. In this case, our coefficient of x is 4, this means the slope of the line is 4. We can prove this by choosing two points on the line and using our slope formula. We know one point would be the y-intercept of (0,20). As another point, we could plug in a 0 for y and solve for x.

0 = 4x + 20

-4x = 20

x = -5

This gives us an ordered pair of (-5,0). Let's plug these two points into our slope formula. We will label (0,20) as point 1 and (-5,0) as point 2: $$m=\frac{0 - 20}{-5 - 0}=\frac{-20}{-5}=4$$ We can see that our slope is 4 either way. Moving forward, we will place an equation in slope-intercept form to obtain the slope since it is a much faster approach.

### Determine if the Slope of a Line is Positive, Negative, Zero, or Undefined

Before we wrap up our lesson, it is important to understand the difference between a line with a positive slope, and a line with a negative slope. Additionally, we will encounter lines with a slope of zero and lines with an undefined slope.#### Positive Slope

A line with a positive slope rises as we move from left to right on our graph. We can picture a line with a positive slope by thinking about a car driving on a road with an incline.#### Negative Slope

A line with a negative slope falls as we move from left to right on our graph. We can picture a line with a negative slope by thinking about a car driving on a road with a decline.#### Slope of Zero

We have already seen that a horizontal line has a slope of zero. We can picture this by thinking about a car driving on a flat road.#### Undefined Slope

We have already seen that a vertical line has an undefined slope. This is due to our calculation of slope. We know that division by zero is undefined. Since we end up dividing by 0 when there is no change on the horizontal axis, we can say a vertical line has an undefined slope. We can picture this as a spaceship flying straight up into space.#### Skills Check:

Example #1

Find the slope. $$(4, 10), (9, -1)$$

Please choose the best answer.

A

$$m=-5$$

B

$$m=4$$

C

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

D

$$m=-\frac{11}{5}$$

E

$$m=-2$$

Example #2

Find the slope. $$(3, -11), (5, 20)$$

Please choose the best answer.

A

$$m=-2$$

B

$$m=-3$$

C

$$m=-\frac{2}{31}$$

D

$$m=\frac{31}{2}$$

E

$$m=\frac{1}{3}$$

Example #3

Find the slope of each line. $$4x - 9y=36$$

Please choose the best answer.

A

$$m=\frac{3}{4}$$

B

$$m=-8$$

C

$$m=-1$$

D

$$m=\frac{4}{9}$$

E

$$m=-2$$

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