**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**

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

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

**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:**

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

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

*slope* = 0

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

*slope* = undefined

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

Given two points and

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).

***Simplify**

**The slope of the line is -1.**

***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.**

***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.**

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.

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:**

***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.**

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.

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.**

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.

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:**

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.**

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.

If you said horizontal, 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:**

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.**

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.**

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

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.**

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. **

***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.**

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)

(answer/discussion to 1a)

1b. (4, 2) and (4, -2)

(answer/discussion to 1b)

(answer/discussion to 1b)

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

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)

(answer/discussion to 3a)

3b.
and

(answer/discussion to 3b)

(answer/discussion to 3b)

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

** Need Extra Help on these Topics?**

**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.**

Last revised on July 3, 2011 by Kim Seward.

All contents copyright (C) 2001 - 2011, WTAMU and Kim Seward. All rights reserved.