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Beginning Algebra
Tutorial 23:Slope


 

deskLearning Objectives


After completing this tutorial, you should be able to:
  1. Find the slope given a graph or two points.
  2. Know the relationship between slopes of parallel lines.
  3. Know the relationship between slopes of perpendicular lines.




desk 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.

 

 

desk 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

positive slope

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


 
Negative slope:

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 x1 and y1 and x2 and y2

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. 

 

 

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


 
example 1

*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.


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

 
example 2
*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.


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

 
example 3
*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.


 

Parallel Lines and Their Slopes
parallel slope

 
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 slope


 

Perpendicular Lines and Their Slopes
perpendicular slope

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

perpendicular slope


 

 

desk 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.

 

pencil Practice Problems 1a - 1d: Find the slope of each line if it exists.

 


 


 

1b.

problem 1b


 

1c.

problem 1c


 

1d.

problem 1d



 

 

pencilPractice Problems 2a - 2b: Find the slope of the straight line that passes through the given points.

 

2a.  (3, 5) and (-1, -8)
(answer/discussion to 2a)

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

 

 

 

desk Need Extra Help on these Topics?


 

The following is a webpage 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.


 

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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Last revised on July 31, 2011 by Kim Seward.
All contents copyright (C) 2001 - 2011, WTAMU and Kim Seward. All rights reserved.