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College Algebra: Introduction to Solving Equations

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About this Lesson

  • Type: Video Tutorial
  • Length: 9:29
  • Media: Video/mp4
  • Use: Watch Online & Download
  • Access Period: Unrestricted
  • Download: MP4 (iPod compatible)
  • Size: 102 MB
  • Posted: 06/26/2009

This lesson is part of the following series:

College Algebra: Full Course (258 lessons, $198.00)
College Algebra: Equations & Inequalities (50 lessons, $65.34)
College Algebra: Linear Equations (8 lessons, $11.88)

Taught by Professor Edward Burger, this lesson was selected from a broader, comprehensive course, College Algebra. This course and others are available from Thinkwell, Inc. The full course can be found athttp://www.thinkwell.com/student/product/collegealgebra. The full course covers equations and inequalities, relations and functions, polynomial and rational functions, exponential and logarithmic functions, systems of equations, conic sections and a variety of other AP algebra, advanced algebra and Algebra II topics.

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 UT-Austin 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, p-adic 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.

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You know, in algebra, a lot of times someone hands us a sort of complicated equation and then we're supposed to somehow solve it. So what exactly does that mean? What does it mean to sort of solve and equation?
Well, first of all, what does an equation even look like? An equation basically has the shape of something like this. So it's a 4x - 3 = 2x + 5. So an equation is basically just a whole bunch of symbols where there's an equal sign. That's where the word "equation" comes from, two things that are being equated. Now, what things are being equated? Well, they're actually numbers, so this could be like 16 = 16. That's what it means. If it was 16 = 15, it wouldn't be a true equation, it would be false. And so you can think of these as a balance scale. So these things weigh perfectly the same amount and they balance perfectly. That's why they're equal.
Okay, well if it was 15 = 15, there wouldn't be much talking about it, because we'd say, "Okay, great, 15 = 15. That's not big news." But, unfortunately, one of the numbers in this equation has been hidden from sight, and that's the x. The x is actually an unknown or a variable. So I don't know what it is, but I do know that, whatever it is, it makes these two things balance out just perfectly. And so, the question is what is the x, what is the number that's actually going to make this thing balance out? And you see, really, if you think about - I don't know if you can hear that or not, but there's a plane going overhead. Anyway, the question is what is the right number to put in here for x? If x bothers you, by the way, just think about it as a question mark. In fact, I can just put a question mark in there, if you want. It's just some mysterious number. You see, I can put a little question mark right there. So it's some question mark thing, and this notation, of course, this means 4 times the question mark. So 4 ?, and then you subtract 3. That's some number. But then if you take the same question mark, that same mysterious number that no on knows and multiply it by 2 and then up the ante by adding 5 to that, those two things have to balance out just perfectly. And the question is what's the question mark? Well, okay, instead of writing question marks, which would look a little funny, we use a letter. But a letter just represents some number that is not known as of yet. It's a variable, it's an unknown.
Now, how do we go about actually figuring out what number should be there? Well, we could try it trial and error. We could say, "Yeah, maybe x = 1." Well, the way to see that would be to replace all the x's by 1. If I did that, here I'd see 4 1, which is just 4, minus 3, which is 1. So on this side I get 1. If I put a 1 in here, I'd see 2 1, which would be 2, plus 5, which would be 7. So I'd have a 1 and a 7. Equal? No, because the 7 would be much heavier. So, in fact, x can't be 1, x must be something else.
Well, you can go through all the numbers, but, unfortunately, there are infinitely many numbers and I don't have that kind of time. So how would you actually figure out what x would work? Well, what you do is you sort of manipulate the equation, you massage the equation. At the end of the day, what's our fantasy? At the end of the day, we'd love to have something that looks like this: x equals, and then the number. So that's what we'd like. So what I want to do is convert this expression, which has the x's all inter-knotted throughout, to an expression where it's just a clean x on the left, an equal sign, and then a number on the right. So that's the goal.
Now, how do you actually achieve that goal? Well, the way you achieve that goal is by just doing some arithmetical steps; that is, you add, subtract, multiply or divide in order to get the x all by itself on the left. So let's think about that for a second. Well, if I want to get all the x's together, I'd like to take this thing and have it be on the other side. But how can I do that? Well, think of it like a balance scale. This equality is like a balance scale. So whatever you do to one side, you've got to do the same thing to the other side, otherwise we'd lose that balance. So if I want to bring this over to this side, what I would want to do is say, "Well, let's see, this is actually being added to 5. So if I subtract off the 2x on both sides, then, in fact, I'd still balance, and then this would be gone, because 2x - 2x = 0." So what I'm going to do is actually subtract off 2x from both sides. Notice that 2x is the same thing as 2x, so I'm not changing anything. The scales remain balanced. But now, notice that on this side I have a 2x - 2x. That actually is zero. And still on this side I have this zero, but this 5 remains. So when I combine, I just have a 5 on this side. That's zero. On this side, what do I have? Well, I have a 4x and I take away two x's. That would leave me with two x's. And this minus 3 would still be there, because I haven't done anything to the 3's.
Well, this is actually looking pretty good. Unfortunately, what I want is x alone. So it's now this -3 that's sort of disturbing. How can I get the -3 to disappear? Well, I can't just remove it, because that would change the value. So what do I do? Well, I think about a way of removing it in a balanced way. So since I'm subtracting 3, my fantasy would be to add 3. So if I add 3 to this side, that would get rid of it. But to keep these scales balanced, I have to add 3 to the other side. So I'm going to add 3 to both sides and what happens? On this side I still have that 2x. That hasn't gone anywhere. And now I have a minus 3 and a plus 3, and they add up to give zero, so that's done. And here I have 5 + 3 = 8. So I've taken this original equation and I've just massaged it a little bit and I got it down to this new equation that represents the same thing.
Now I want x alone, and I see that 2 is multiplying the x, so I want to get rid of that 2. Now maybe a good idea is to subtract 2 from both sides. Well that would be a great idea, except the 2 is not hooked up to the x with an addition, but instead a multiplication. And the thing that undoes multiplication is division. So instead of subtracting 2 to both sides, I'd want to divide both sides by 2, thus canceling out that multiplying 2 in front. So if I divide both sides by 2, then on this side just cancels out and I just have 1. So this just gives me an x. And on this side I have , which gives me a 4.
And so I finally see that x = 4. And this is another equation, but it's an equation that's very easy to read. It says x = 4. So that's the mystery. And we can actually go back and replace the x by 4 and make sure this really works. If I put a 4 in here, that would be 4 4, which is 16, 16 - 3 = 13. So this side I'm weighing 13. What am I weighing on this side? If I put a 4 in here, I see 2 4, which would be 8, 8 + 5 = 13. Notice they balance out perfectly. If you put any other value in for x, they would not balance out. This is the only number that makes this thing balance out. There's only one answer to this question of what is the x value for this. It turns out the answer is 4.
Now, when you look at what we did here, really what you can see is that, in some sense, instead of just thinking about subtracting both sides and adding things to both sides and so forth, you could visualize it, if you wanted to, as sort of just moving terms around. I mean, think about this for a second. I had this 2x here and my fantasy was to get rid of it and push it onto the other side. But in order for me to move it to the other side, I actually had to change the sign, because I had to subtract it to make sure it goes away. So really what's going on, by the way, the math part is I'm subtracting it from both sides. Here, I was adding 3 to both sides. But the non-math part, you could think about it like this: I just want to take this part here - if you want to take something and bring it to the other side that's being added, all you've got to do is bring it over to the other side, but subtract it. If you just think about it that way, if I bring that over and subtract it, there it is. I subtract it and then I get this value.
Now, what about this? Well, here I'm subtracting a 3. If I want to bring that to the other side, what I would do is change the sign, so now I'd add it. That's exactly what happened. See, this goes away, so it's just now the 2x. And here it comes out here and becomes an 8. And then I just divided both sides by the 2, in order to get rid of that initial factor there, and I get the answer.
So you could actually think about solving an equation by moving terms back and forth that are being added or subtracted by doing the opposite. So if it's being added, you'd subtract it, if it's being subtracted, you add it to the other side. Or if you have a multiple in front, you have to divide both sides by that thing to cancel them out. Just like if you had a division here, you would multiply through by both sides, in order to cancel that out.
So that's sort of the nuts and bolts of solving linear equations. You want to find the variable, you want to find the unknown. Don't let that freak you out. Don't let the x freak you out, it's just something we don't know. Let's find it. That's how you find it.
Equations and Inequalities
Linear Equations
An Introduction to Solving Equations Page [1 of 2]

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