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Find the mistake: two-step equations

When doing algebra, we sometimes make mistakes! It's important to be able to check our own work (and other people's work) for correctness.

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Video transcript

- [Voiceover] What I wanna do in this video is get some practice building an important skill, being able to look at some algebra that has happened and see whether it is correct. And this is valuable. You might look at other people's' work, you might look at your own work and be able to tell, "Well, hey, did I do that right?" The logic maybe broke down. Or why am I not getting the answer that I think I should get? So let's get some practice here. So I have an equation, nine times, and then in parentheses, two x plus 1/3 is equal to 39. And then someone has worked it out. This is step one, step two, then step three. They get to this solution, x equals two. What I want you to do is pause this video and figure out, is this solution correct? Did all of the steps they do, do they make logical sense, do they make algebraic sense? Or did they make a mistake in one of these steps, and if so, which of these steps did they make a mistake in? All right, so let's just work through this problem and see if all of these steps are reasonable. So in this first step, let's see, they got an 18x here, so it looks like they distributed the nine. So nine times two x would be 18x, and then of course, you would also have to do nine times 1/3. Nine times 1/3 would be 9/3, or three, so that seems reasonable. And you still have a 39 here on the right-hand side. So step one checks out. Now, let's see, what did they do over here? After step one, or at this stage, you had 18x plus three, and then over here you only have 18x. So they must've subtracted three from the left-hand side to get rid of this three. So they subtracted three from the left-hand side. Well, if they subtract three from the left-hand side, they also need to subtract three from the right-hand side, and it looks like they did. 39 minus three is 36. So that step checks out. And so you have 18x is equal to 36. And so to solve for x, you would just wanna divide both sides by 18. So you would get x is equal to 36 over 18, or 2. So all of these steps check out, and so there's no problem here, no mistake. And you can verify that x does indeed equal two. Let's do some more questions, or see if we can identify more errors or maybe no errors. All right, we have 8/3 is equal to three times c plus 5/3, and we wanna solve for c. Pause the video and see if any of these mistakes are, or if any of these steps, are incorrect or maybe there's no mistake at all. All right, let's go step by step. So in step one, kept the left-hand side the same. The right-hand side, let's see, you go from three times this to three c, so it looks like they're distributing the three. So three times c is indeed three c, and then you wanna distribute three times 5/3, well, that's gonna be five. That's not gonna be 5/3. This thing right over here, this thing needs to be five, not 5/3, so that is not correct. So step one is where we have the problem. Let's keep going. Let's do one more of these. I find this strangely entertaining. All right, this is a little bit simpler. 1/4 r plus two is equal to 10. And then they immediately go from that to just 1/4 r, so they got rid of this two on the left-hand side. Well, to do that you could subtract two from the left, but, of course, you would also have to do that from the right-hand side. And so you get 1/4 r plus two minus two, that would be just 1/4 r. So the left-hand side makes sense. 10 minus two is not 12. 10 minus two is eight. This right over here should be eight. They added two on the right-hand side. They subtracted two on the left-hand side, and they added two on the right-hand side. Then the equality wouldn't hold anymore. So you definitely have an error. Step one is not correct.