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
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:
Note that when a line has a positive slope it goes up
left to right. 
Negative slope:
Note that when a line has a negative slope it goes
down left to right. 
Zero slope:
slope = 0
Note that when a line is horizontal the slope is 0. 
Undefined slope:
slope = undefined
Note that when the line is vertical the slope is
undefined. 
Slope Formula Given Two Points
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). 

*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 nonzero 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 yintercept.
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 yintercept. 
Example
4: Find the slope and the yintercept
of the line . 
As mentioned above, if the equation is in the slope/intercept
form,
we can easily see what the slope and yintercept
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 yintercept 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 yintercept
is b, which is the constant. In our problem, that would
be
2.
The answer is the slope is 1 and the yintercept is
2. 
Example
5: Find the slope and the yintercept
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 yintercept. 

*Written in slope/intercept form

Lining up the form with the equation we got, can you
see what the slope
and yintercept 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 yintercept
is b, which is the constant. In our problem, that would
be
1.
The answer is the slope is 2 and the yintercept is
1. 
Example
6: Find the slope and the yintercept
of the line x = 5. 
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 yintercept.
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 yintercept.
Looking at the graph, you can see that this graph never crosses the yaxis,
therefore there is no yintercept
either.
Another way to look at this is the x value
has to be 0 when looking for the yintercept
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 yintercept does not
exist. 
Example
7: Find the slope and the yintercept
of the line y =
2. 
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 yintercept.
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 nonzero number in the denominator means only
one thing. The slope equals 0.
Now let’s look at the yintercept.
Looking at the graph, you can see that this graph crosses the yaxis
at (0, 2). So the yintercept is (0, 2).
The slope is 0 and the yintercept
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 yintercepts.

Perpendicular Lines and Their
Slopes

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

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 yintercepts
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.
Practice
Problems 2a  2c: Find the slope and the yintercept
of the line.
Practice
Problems 3a  3b: Determine if the lines are parallel,
perpendicular,
or neither.
Practice
Problem 4a: Determine the slope of the line.
Need Extra Help on these Topics?
Last revised on July 3, 2011 by Kim Seward.
All contents copyright (C) 2001  2011, WTAMU and Kim Seward. All rights reserved.
