Intermediate Algebra
Tutorial 15: The Slope of a Line

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Learning Objectives

After completing this tutorial, you should be able to:

Find the slope given a graph, two points or an equation.
Write a linear equation in slope/intercept form.
Determine if two lines are parallel, perpendicular, or neither.

Introduction

This tutorial takes us a little deeper into linear equations. We will be looking at the slope of a line. We will also look at the relationship between the slopes of parallel lines as well as perpendicular lines. Let's see what you can do with slopes.

Tutorial

Slope

The slope of a line measures the steepness of the line.

Most of you are probably familiar with associating slope with "rise over run".

Rise means how many units you move up or down from point to point. On the graph that would be a change in the y values.

Run means how far left or right you move from point to point. On the graph, that would mean a change of x values.

Here are some visuals to help you with this definition:

Positive slope:

positive slope

Note that when a line has a positive slope it goes up left to right.

Negative slope:

negative slope

Note that when a line has a negative slope it goes down left to right.

Zero slope:

zero slope

slope = 0

Note that when a line is horizontal the slope is 0.

Undefined slope:

undefined slope

slope = undefined

Note that when the line is vertical the slope is undefined.

Slope Formula Given Two Points

Given two points and

slope formula

The subscripts just indicate that these are two different points. It doesn't matter which one you call point 1 and which one you call point 2 as long as you are consistent throughout that problem.

Note that we use the letter m to represent slope.

Example 1: Find the slope of the straight line that passes through (-5, 2) and (4, -7).

*Plug in x and y values into slope formula

*Simplify

Make sure that you are careful when one of your values is negative and you have to subtract it as we did in line 2. 4 - (-5) is not the same as 4 - 5.

The slope of the line is -1.

Example 2: Find the slope of the straight line that passes through (1, 1) and (5, 1).

*Plug in x and y values into slope formula

*Simplify

It is ok to have a 0 in the numerator. Remember that 0 divided by any non-zero number is 0.

The slope of the line is 0.

Example 3: Find the slope of the straight line that passes through (3, 4) and (3, 6).

*Plug in x and y values into slope formula

*Simplify

Since we did not have a change in the x values, the denominator of our slope became 0. This means that we have an undefined slope. If you were to graph the line, it would be a vertical line, as shown above.

The slope of the line is undefined.

Slope/Intercept Equation of a Line

If your linear equation is written in this form, m represents the slope and b represents the y-intercept.

This form can be handy if you need to find the slope of a line given the equation.

Function Notation of the
Slope/Intercept Equation of a Line

m still represents slope and b still represents the y-intercept.

Example 4: Find the slope and the y-intercept of the line

As mentioned above, if the equation is in the slope/intercept form, we can easily see what the slope and y-intercept are.

Let’s go ahead and get it into the slope/intercept form first:

*Sub. 3x and add 6 to both sides

*Inverse of mult. by 3 is div. by 3

*Written in slope/intercept form

Lining up the form with the equation we got, can you see what the slope and y-intercept are?

In this form, the slope is m, which is the number in front of x. In our problem, that would have to be -1.

In this form, the y-intercept is b, which is the constant. In our problem, that would be 2.

The answer is the slope is -1 and the y-intercept is 2.

Example 5: Find the slope and the y-intercept of the line

This example is written in function notation, but is still linear. As shown above, you can still read off the slope and intercept from this way of writing it.

In this example, it is already written in the slope/intercept form, so we do not have to mess around with it. We can get down to business and answer our question of what are the slope and y-intercept.

*Written in slope/intercept form

Lining up the form with the equation we got, can you see what the slope and y-intercept are?

In this form, the slope is m, which is the number in front of x. In our problem, that would have to be 2.

In this form, the y-intercept is b, which is the constant. In our problem, that would be -1.

The answer is the slope is 2 and the y-intercept is -1.

Example 6: Find the slope and the y-intercept of the line x = 5.

Note how we do not have a y. This type of linear equation was shown in Tutorial 14: Graphing Linear Equations. When we have x = c, where c is a constant, then this graph is what type of line?
If you said vertical, you are correct.

Since this is a special type of linear equation that can’t be written in the slope/intercept form, I’m going to give you a visual of what is happening and then from that let’s see if we can’t figure out the slope and y-intercept.

The graph would look like this:

example 6a

First, let’s talk about the slope. Note that all the x values on this graph are 5. That means the change in x, which is the denominator of the slope formula, would be 5 - 5 = 0. Well you know that having a 0 in the denominator is a big no, no. This means the slope is undefined. As shown above, whenever you have a vertical line your slope is undefined.

Now let’s look at the y-intercept. Looking at the graph, you can see that this graph never crosses the y-axis, therefore there is no y-intercept either. Another way to look at this is the x value has to be 0 when looking for the y-intercept and in this problem x is always 5.

So, for all our efforts on this problem, we find that the slope is undefined and the y-intercept does not exist.

Example 7: Find the slope and the y-intercept of the line y = -2.

Note how we do not have an x. This type of linear equation was shown in Tutorial 14: Graphing Linear Equations. When we have y = c, where c is a constant, then this graph is what type of line?
If you said horizontal, you are correct.

The graph would look like this:

example 7a

First, let’s talk about the slope. Note how all of the y values on this graph are -2. That means the change in y, which is the numerator of the slope formula would be -2 - (-2) = 0. Having 0 in the numerator and a non-zero number in the denominator means only one thing. The slope equals 0.

Now let’s look at the y-intercept. Looking at the graph, you can see that this graph crosses the y-axis at (0, -2). So the y-intercept is (0, -2).

The slope is 0 and the y-intercept is -2.

Parallel Lines and Their Slopes

In other words, the slopes of parallel lines are equal.

Note that two lines are parallel if there slopes are equal and they have different y-intercepts.

parallel lines

Perpendicular Lines and Their Slopes

In other words, perpendicular slopes are negative reciprocals of each other.

perpendicular lines

Example 8: Determine if the lines are parallel, perpendicular, or neither.

and

In order for these lines to be parallel their slopes would have to be equal and to be perpendicular they would have to be negative reciprocals of each other.

So let’s find out what the slopes are. Since the equations are already in the slope/intercept form, we can look at them and see the relationship between the slopes. What do you think? The slope of the first equation is 7 and the slope of the second equation is 7.

Since the two slopes are equal and their y-intercepts are different, the two lines would have to be parallel.

Example 9: Determine if the lines are parallel, perpendicular, or neither.

and

Again, the equations are already in the slope/intercept form, so let’s go right to looking for the slope. What did you find?

I found that the slope of the first equation is 4 and the slope of the second equation is -1/4. So what does that mean?

Since the two slopes are negative reciprocals of each other, the two lines would be perpendicular to each other.

Example 10: Determine if the lines are parallel, perpendicular, or neither.

and

Writing the first equation in the slope/intercept form we get:

*Inverse of add 10x is sub. 10x
*Written in slope/intercept form

Writing the second equation in the slope/intercept form we get:

*Inverse of add 4x is sub. 4x

*Inverse of mult. by 2 is div. by 2

*Written in slope/intercept form

In order for these lines to be parallel their slopes would have to be equal and to be perpendicular they would have to be negative reciprocals of each other. So let’s find out what the slopes are. Since the equations are now in the slope/intercept form, we can look at them and see the relationship between the slopes. What do you think?

The slope of the first equation is -10 and the slope of the second equation is -2.

Since the two slopes are not equal and are not negative reciprocals of each other, then the answer would be neither.

Practice Problems

These are practice problems to help bring you to the next level. It will allow you to check and see if you have an understanding of these types of problems. Math works just like anything else, if you want to get good at it, then you need to practice it. Even the best athletes and musicians had help along the way and lots of practice, practice, practice, to get good at their sport or instrument. In fact there is no such thing as too much practice.

To get the most out of these, you should work the problem out on your own and then check your answer by clicking on the link for the answer/discussion for that problem. At the link you will find the answer as well as any steps that went into finding that answer.

Practice Problem 1a - 1b: Find the slope of the straight line that passes through the given points.

1a. (3, 5) and (-1, -8)
(answer/discussion to 1a)

1b. (4, 2) and (4, -2)
(answer/discussion to 1b)

Practice Problems 2a - 2c: Find the slope and the y-intercept of the line.

2a.
(answer/discussion to 2a)

2b. x = -2
(answer/discussion to 2b)

2c. y = -1
(answer/discussion to 2c)

Practice Problems 3a - 3b: Determine if the lines are parallel, perpendicular, or neither.

3a.

and

(answer/discussion to 3a)

3b.

and

(answer/discussion to 3b)

Practice Problem 4a: Determine the slope of the line.

4a.

(answer/discussion to 4a)

Need Extra Help on these Topics?

The following are webpages that can assist you in the topics that were covered on this page:

http://www.purplemath.com/modules/slope.htm
This webpage helps you with slope.

http://www.math.com/school/subject2/lessons/S2U4L2DP.html
This website covers slopes and y-intercept.

Go to Get Help Outside the Classroom found in Tutorial 1: How to Succeed in a Math Class for some more suggestions.

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