Learning Objectives
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
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
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
The slope of the line is 0.
*Simplify
The slope of the line is undefined.
Slope/Intercept Equation of a LineThis form can be handy if you need to find the slope of a line given the equation.
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
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.
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.
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.
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:
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.
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:
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 SlopesNote that two lines are parallel if there slopes are equal and they have different y-intercepts.
Perpendicular Lines and Their SlopesSo 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.
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
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.
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 y-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.
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.
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