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Pre-Algebra: Writing & Solving Two-Step Equations

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About this Lesson

  • Type: Video Tutorial
  • Length: 5:09
  • Media: Video/mp4
  • Use: Watch Online & Download
  • Access Period: Unrestricted
  • Download: MP4 (iPod compatible)
  • Size: 55 MB
  • Posted: 01/28/2009

This lesson is part of the following series:

Pre-Algebra Review (31 lessons, $61.38)

This lesson outlines the process for solving two step equations, which are equations that involve more than one operation to solve for the variable. First, Professor Burger teaches you to isolate the variable terms. Remember that you will use the reverse order of operations to solve an equation with more than one operation involved. Once you have isolated the variable term on one side of the equation, you can solve for it.

Taught by Professor Edward Burger, this lesson was selected from a broader, comprehensive course, Pre Algebra. This course and others are available from Thinkwell, Inc. The full course can be found at http://www.thinkwell.com/student/product/prealgebra. The full course covers whole numbers, integers, fractions and decimals, variables, expressions, equations and a variety of other pre algebra 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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Thinkwell
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Founded in 1997, Thinkwell has succeeded in creating "next-generation" textbooks that help students learn and teachers teach. Capitalizing on the power of new technology, Thinkwell products prepare students more effectively for their coursework than any printed textbook can. Thinkwell has assembled a group of talented industry professionals who have shaped the company into the leading provider of technology-based textbooks. For more information about Thinkwell, please visit www.thinkwell.com or visit Thinkwell's Video Lesson Store at http://thinkwell.mindbites.com/.

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SOLVING A TWO-STEP EQUATION

We’ve been solving one-step equations; we know how to do that. We just undo whatever the operation is. It turns out that it doesn’t get any harder if we have two different operations going on. Those are called two-step equations. Not a big deal, let’s everyone relax.

Let’s take a look at an example and see that the key is to try to isolate the unknown first. Here I see 3n plus 1 equals 19. It’s a two-stepper, because I’m both multiplying and adding. Now, what do we do? The thing I want to do, the goal at the end of the day, is to have n equals bang. Okay? n equals bang. So how do I do that? I’ve got to get n by itself. And the first thing I want to do is get all the numbers that are n-free to the other side. So, for example, I have a plus 1 here; I don’t like that. So, first I want to deal with that, and then I’ll deal with the 3 out in front of the n. First, let’s isolate and get all the like terms over.

I have 3n plus 1, what do I do? I no longer want to have that plus 1, so I want to subtract 1 from this side. But to keep everything balanced perfectly, I have to subtract 1 from the right-hand side. Remember if I do one thing to one side, and one thing to the other side, it’s still an equivalent equation. So if I subtract 1, what do I see? On the left I see 1 minus 1 is 0, so I’m just left with that 3n, which is exactly what I was after, 3n. Equals, and then 19 minus 1 is 18.

Now I’m down to a one-step equation, which we know how to solve. I want to get rid of the 3. I want to undo the multiplication of 3 by dividing both sides by 3. If I do it on this side, I have to do it on this side. I see n equals 6. That’s it.

Let’s try another one together. Check this one out. 21 equals negative 2p minus 5. Maybe you’re not happy with having the p on the right-hand side? It doesn’t bother me, but if you don’t like it you can always bring it over to the left-hand side if you want. The key thing to do is to have all the p’s together and all the numbers without p’s on the other side, whatever the other side is. This is definitely a two-stepper, because I’m multiplying p by something and then I’m taking that and subtracting 5, so I’ve got subtraction and multiplication, two steps.

What to do first? I want to isolate the p, so I want to get the non-p’s to the other side. I’m going to keep p on the right just to show you that I’m not afraid of that. Now I want to undo the subtracting of 5, so I’m going to add 5, 5 minus 5 would give me 0, but I have to do that to both sides to make sure this thing balances out and I get an equivalent equation. I’m going to add 5 to both sides. This won’t change anything. And, in this case, I’m going to get 26 equals negative 2p, and this becomes a 0.

Notice I’ve reduced this to a one-step equation, right? Because now all I have to do is undo the multiplication of negative 2, which I do by dividing by negative 2. You divide both sides by negative 2, and I see p equals, and 26 divided by negative 2, a positive divided by a negative is a negative, 26 divided by 2 is 13, so I see negative 13 is the answer. By the way, you might want to write that as p equals negative 13, it’s the same thing. You can write an equality either way. And what you want to do now is to check your answer to make sure that this answer makes sense.

What can you do? Wherever you see a p, you can plug in negative 13. Let’s check it, here’s the check. And of course if you check you must use green. On the one hand I’ve got 21, on the other hand I’ve got negative 2 p minus 5. If I plug in negative 13, negative 2 times negative 13 minus 5, hmmm. A negative times a negative is a positive, 2 times 13 is 26, minus 5. 26 minus 5 is 21, and we have a winner. 21 equals 21, it checks. And you can see the answer is correct. Cool.

Solving two-step equations, not a big deal. You try to isolate the unknown, get everything else to the other side, and then undo the multiplication or division and you’re all good to go.

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