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About this Lesson
 Type: Video Tutorial
 Length: 12:35
 Media: Video/mp4
 Use: Watch Online & Download
 Access Period: Unrestricted
 Download: MP4 (iPod compatible)
 Size: 135 MB
 Posted: 12/02/2008
This lesson is part of the following series:
College Algebra: Full Course (258 lessons, $198.00)
Trigonometry: Full Course (152 lessons, $148.50)
College Algebra: Relations and Functions (57 lessons, $74.25)
Trigonometry: Algebra Prerequisites (60 lessons, $69.30)
Beginning Algebra Review (19 lessons, $37.62)
College Algebra: The Slope of a Line (2 lessons, $2.97)
In this lesson, you will learn how to find the slope, or the relative increase (also known as a pitch), of a line if you are given two points on that line (x1, y1) and (x2, y2). The slope (denoted by the letter, m) of a line is defined by the change in y divided by the change in x. First, you must calculate the change in the two distances (or the change (x1  x2) and the change (y1  y2)). You will also learn the shorthand for writing the equation of a slope and the phrase 'rise over run.' After learning how to find the slope of a line, you will practice with several sets of points and lines with different slopes (including verticle lines and horizontal lines) and also practice graphing those lines to view the slope. Professor Burger also examines what it means when a slope is undefined (or the change in x = 0), and when a slope = 0 (or the change in y = 0).
Taught by Professor Edward Burger, this lesson was selected from a broader, comprehensive course, Beginning Algebra. This course and others are available from Thinkwell, Inc. The full course can be found at http://www.thinkwell.com/student/product/beginningalgebra. The full course covers linear equations, inequalities, polynomials, rational expressions, relations and functions, roots and radicals, quadratic equations and systems of equations.
Edward Burger, Professor of Mathematics at Williams College, earned his Ph.D. at the University of Texas at Austin, having graduated summa cum laude with distinction in mathematics from Connecticut College.
He has also taught at UTAustin and the University of Colorado at Boulder, and he served as a fellow at the University of Waterloo in Canada and at Macquarie University in Australia. Prof. Burger has won many awards, including the 2001 Haimo Award for Distinguished Teaching of Mathematics, the 2004 Chauvenet Prize, and the 2006 Lester R. Ford Award, all from the Mathematical Association of America. In 2006, Reader's Digest named him in the "100 Best of America".
Prof. Burger is the author of over 50 articles, videos, and books, including the trade book, Coincidences, Chaos, and All That Math Jazz: Making Light of Weighty Ideas and of the textbook The Heart of Mathematics: An Invitation to Effective Thinking. He also speaks frequently to professional and public audiences, referees professional journals, and publishes articles in leading math journals, including The Journal of Number Theory and American Mathematical Monthly. His areas of specialty include number theory, Diophantine approximation, padic analysis, the geometry of numbers, and the theory of continued fractions.
Prof. Burger's unique sense of humor and his teaching expertise combine to make him the ideal presenter of Thinkwell's entertaining and informative video lectures.
About this Author
 Thinkwell
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 Joined:
11/13/2008
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Thinkwell lessons feature a starstudded cast of outstanding university professors: Edward Burger (PreAlgebra through...
More..Recent Reviews
 I always forget how to do this! Got it now, th...
 06/04/2009

I can never quite remember how to calculate slope once i have two different points on the line, and then, depending on what other information I have, I run into problems generating the equation for the line. This really helped me understand not only how to use the formula but also how to think about it and why it works (and thus, hopefully, I'll remember it from now on).
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FINDING THE SLOPE GIVEN TWO POINTS
So, how would we measure this pitch of a line, which could be not very steep at all or very, very steep. Well, basically to measure steepness we just want to see the rate of change in some sense. Is it a very steep change, steep grade or a light grade? The way we do that is by taking two points that are on the line and measuring their relative increase from the x and from the y. Basically, if I have two points, this is now what we call slope, the slope of the line. How do you find it? The way you find it is by taking two points that are on the linelet’s suppose that we have this line. We take two points. Maybe I’ll call this one the coordinates of this (x1,y1) and the points of this, let me call the coordinates (x2,y2). So those are the coordinates of that point.
Then if I want to find the slope, I want to compute that pitchwhat I do is, I see sort of how far I traveled in this direction and then I look how far I traveled in this direction. I’d compare those things. Now this length is just the change in the x. So this length is quite literally x1  x2. It’s just the change in the x’s, how far I’ve traveled. Now to find this distance, I just subtract the exact same way. I’ll say y1 y2. That represents the displacement in this direction. If I want to weight these things and see how they compare to each other an how they balance off, well I define the slope, so the slope will equal the change in y divided by the change in x. I would see y1  y2 all divided by x1  x2. That’s the definition of the slope of a line and this is what determines and measures steepness, slope.
There’s a shorthand way of writing that. Let me just show you the shorthand way. Sometimes, by the way, in fact a lot of times, people refer to slope of a line with one letter, instead of writing slope each time. What do you think the letter is? Well obviously it’s m, m for slope. Well, anyway, that’s standard, so I’m going to use that here. A little m actually represents the slope. So, I could write m here, m equals y1  y2 divided by x1  x2, but there’s even a shorthand name for this. The shorthand name for this is actually just to capture the idea of the change in y over the change in x. So, sometimes we use the Greek letter ?, which is the triangle, for change.
So, this looks really complicated, but don’t panic at all. This is not a variable. This just represent the change in y divided by the change in x. Literally, it’s this, but it’s a shorthand way of writing that. The change in y over the change in x. Sometimes people use the little phrase rise over run. The idea is you take the rise part, that’s the change in y and you put it over the run, how much you’re running in the x direction. Sometimes people remember slope as saying, “slope is rise over run.” Some people remember it as change in y divided by change in x. Other people remember it as the slope is just y1  y2 divided by x1  x2, where (x1,y1) and (x2,y2) are two different points on the line.
Given that, let’s actually compute some slopes. Let’s find the slopes between some points here. So, let’s suppose that we’re given the following points. The question is always, “What is the slope of the line that passes through these two points?” The first points, (1,1) and (5,5). What’s the slope of the line that passes through these two points? Well, how do you find them? Well, we take the y differences and then the x differences. So, the slope here would be what? This would be 5  1 divided byso that’s the y minus the y. There they are. Then divided by the x minus the x. So that would be still 5  1. This equals 4 over 4, which equals 1. So, the slope of this line is 1. The slope of this line equals 1. What does that look like graphically?
Let me show you on this grid. I just want to mark the axis here with this big red dot. That big red dot represents (0,0). So this is the xaxis right here and that’s the yaxis. It’s hard to detect them because I have all these grid lines in here. That’s okay, that’s the (0,0). Now, let’s see where these points are. I’ve got the point (1,1) and then I have the point (5,5). So there’s the line. It goes through (1,1) and then (5,5). You can see it and that slope is 1. Look what that looks like. It sort of goes right between the axis. If I put these colored things here for the axis, so you can see the axis a little better, you can see that if a slope is 1, it goes right in between, perfectly in between, diagonal between those two axis. That’s slope 1.
Let’s take a look at another example. The next two points that I want to find the slope between, (2,0) and (3,7). What do I do? What I do now is I take rise over run, the change in y’s over the change in x’s. It’s very important here to always subtract this y from that y and then this x from that x. Always subtract the same point from here and then start with the same point from here. So, if I take 7 and subtract 0, then I have to divide that by taking the 3, you see how I always start with the same point, 2. So, I took 7, subtract the 0. I then, did not take 2 minus 3, I took 3 too. If it took 7 here, I take 3 too. If you would have taken 0  (7) then you would have taken 2  (3). So, always subtract the same point from
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the other same point. If you compute that, that equals 7 divided byand this would be a 5, which should be 7/5. So, the slope of this equals 7/5.
What does that mean? What does that look like? Well, we already saw what the slope of 1 looks like. What does slope 7/5 look like? Well, again, here’s the origin, but a slope 7/5 means is that 7/5, it means that I have to go up 7 and then over 5. Rise over run. So, up 7 and then I go over 5, 7/5 would have a slope that looks something like this. So that’s that slope. Which if you remember, the other slope of 1 was right between, right exactly between these to things, right along the diagonal. So, 7/5 is actually steeper. Notice that 7/5 is actually a number bigger than 1. So, what we’re seeing is the bigger the slope the more steep the line. There’s that example.
Let’s take a look at another example. (2,4) and (2,7), let’s compute the slope there, (2,4), (2,7). Let’s now, just for a practice, I’m going to take this y value and subtract that y value. Then, of course, I’m going to take this x value. Doesn’t make a difference what order you take. So, 4  7, but then I have to divide that by 2  2. Well, that’s 3 over 0. Uhoh, something over 0 is undefined. This is undefined, so the slope here equals undefined. What does it mean for slope to be undefined? What would that look like? We have a 0 on the bottom; it means there’s no change in the x. No change in x, so what were those points again? One point was (2,4) and the other point was (2,7). If I connect those two points with a line, it looks like this. No change in x. All the x’s are always equal to 2. So we have a 0 in the denominator. So, if a slope is undefined, what that means is the line must exactly be vertical. So vertical lines have undefined slopes, because there’s no change in x, you have a 0 in the denominator. So, someone says, “I’m thinking of a line and it has a slope that’s undefined.” You know it’s vertical. If someone says to you, “I have a vertical line.” You know it means slope is undefined. So, if undefined, we know it’s vertical.
Let’s try one last one. The last one I want to look at is the following. How about (3,1) as one point and the second point is (1,1). What would be the slope there? Well, we have to subtract away and see what happens. If I subtract 1 from 1, 1  1, then I’ll take 1  (3). Now, don’t write this, because that’s wrong. I have to take 1  (3). Which actually means the top is 0, but the bottom would actually be 1 + 3, which is 4. So, I have 0 over 4, which is defined. It equals 0. So, in fact here, the slope is 0. The slope is 0. What does that mean? What does it mean for slope to be 0? Well, slope to be 0 means there must be no change in the y’s. So what would that look like? Well, if we had (3,1), that’s negative 3, up 1, right there. The other point is (1,1), 1 over, one up. If I connect those things, you see there’s no change in the y’s. The line is always just one unit above the xaxis. That’s a horizontal line. If you think about it a horizontal line has no change in y. So we have a 0 in the top, in the numerator of the slope formula. What we see is horizontal lines have slope 0, no pitch at all. 0 means it’s not pitched anyway.
If it’s vertical that forces the slope to be undefined. Anything in between we see now, the slope will be either 1, which is just like that, or if it’s bigger than 1, it’s like this. If it’s less than 1, it’s like this. What happens with negative slopes? Negative slopes are just the exact same thing, but that means that they’re downward sloping rather than upward sloping. Okay, we’ll take a look at more linear issues coming up next.
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I can never quite remember how to calculate slope once i have two different points on the line, and then, depending on what other information I have, I run into problems generating the equation for the line. This really helped me understand not only how to use the formula but also how to think about it and why it works (and thus, hopefully, I'll remember it from now on).