College Algebra: Equivalent, No Solution Equations

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So constantly in mathematics we're given a question and the idea is can we convert it to something that's equivalent but easier to solve. It's the idea of breaking things down to smaller pieces. That's what I want to talk to you about now, the idea of taking an equation and trying to solve it by finding equivalent equations, and leading that always to the goal of x equals boom answer. Now, when we're faced with a question, it's going to be stuff that has x's all tangled up with equal signs, and we have to clean it up, and it's that x equals boom answer. So this is really the idea of just solving some equations. And really the equations I'm going to be looking at are going to be actually linear equations.
So let's just look at an example here, and you can start to see the idea of trying to write an equation in an equivalent form. So x + 4 = 6. By the way, it's never a bad idea to, when you're given, faced with an issue, to think, "Let me just think about this for a second." Okay, look at that, and maybe you can even guess an answer. I think a good guess might be, "Well, let's see, if I add 2 to 4, I get 6, and then the answer is 2." So it's always good, if you can, to have a guess. Now this is actually not that hard of a question, but when they get harder, it's always good to have a guess, just to see if what you're getting make sense.
Let me give you an example. Suppose you do a lot of work on this question - this is pretty easy, but imagine one that's more complicated - and at the end of the day you've got x = -12. Is it possible here that the answer here could be negative? Now if you just think about that for a second, you can actually see it doesn't make a lot of sense. How do I take something that's smaller than 6, and then subtract stuff, -12, and now it's going to get all the way up to 6; it's impossible. So thinking about if an answer makes sense or not in advance is actually not a bad idea. And making a guess is actually not bad either.
Anyway, that's all; it has nothing to do with what I want to tell you about now, but it's a great lesson for life. What I want to tell you about now is thinking about this now as a balance scale. I want to isolate the x, so I'm going to take this +4, and I want to think of it as migrating it to the side. The way you do that legally is by subtracting 4. But if I subtract 4 on one side, that would now kill the scale, kill the scales of justice. To make sure I smooth things out, I've got to subtract 4 from this side. Now 4 minus -4 equals -4, so if I had an equation, I know it's true, and I do this to it, nothing changes. But here I see x, and then these are the additive inverses, so it's just x plus zero. So x alone perfect equals 2. Voila, just what we thought. So you can see a very simple example of solving by just adding on both sides by the additive inverse.
Here's a different question. How about 30y = 120? Now, by the way, y, oh, maybe we'll have to use a different procedure. Remember letters are just representing things that we up to this point do not know. So don't be at all bothered by a y versus an x versus a w versus a z versus a cough. I'm going to cough for you right now. Don't be confused by any of that. It's all the same thing; it's just an unknown quantity. If you want to solve this for y, well, there's no stuff being added on this side. But I have 30 times y; so to undo a multiplication, the opposite of multiplication is division. So I should divide, and what should I divide both sides by? Well, I want to have y by itself so I want to get rid of all those 30s, so I divide this all by 30. But right now that's not a balanced scale. I've got to scale it out and balance it by doing the same thing on the right. Now the 30s cancel, and I'm just left with y alone, perfect. And here, when you simplify this, you actually get 4, so y equals 4. And, again, you can check your answer. Plug 4 in for y. Is 30 times 4 120? Yes, so it checks. You can always see if your answer is right or not by checking, not a big deal.
Let's try another one together. How about this one, here's a combo: 5x + 6 = 26. So you have a lot of stuff going on. What do I do first? Well, I'm trying to isolate the X. So I'll take all the terms that have no x in it, all those naked terms, and I'll push them over to the other side first. So I just have x's; get all the x's together in one big clump, and then divide through whatever's being multiplied by the x. So the first step is to get rid of that 6. So I subtract 6 on both sides, so it balances, and then I have 5x; that's good, the x's now are just all alone, and here I have 20. Now what do I do? Well, now what I have to do is get rid of the 5. So now I have to divide both sides by 5, and then I see x = because the fives cancel = equals four. You can check your answer. The make the unknown four and see if this is really now going to be true. You put in four hear, this should now become an identity. Let's see: 5 x 4 is 20 + 6 = 26; it's correct, you can check your answer.
Great, let's try another one. How about this: 2x + 5 = x + 2 + x. That's sort of a complicated looking one; notice that there's x's everywhere. So the first step is try to isolate all the x's and put them together. So what I'll do here is I'll write x plus 5 - I won't touch that side, I'll do things slowly - x plus x - how many x's do I have, I have one x, and then I add another x, I now have two x's. So now I have 2x, combining these, and then don't forget the plus 2. Well, now, you might want to bring the x's all to the same side, so why don't I take these two x's and subtract it from this side in order to push it over there. So if I subtract it from that side, what I see is -2x equals - I have to do it on both sides - and the whole point is that here they cancel out, I'm left with zero, and I'm just left with therefore 2 equals - but notice here I have 2x minus 2x, and they cancel out here as well. I wasn't expecting that, but that turns out to be the nature of the question, and it's left with the 5. So this expression, this equality, is equivalent to the equality 5 = 2. Well, that's false, that is just out and out false. That means there are no solutions to this - no solutions - and some people say that this equation is a contradiction, because there is no unknown that you can put in here to make this thing true, because no matter what you put in here, you're going to say that 5 = 2, and that's just not going to happen, it's not going to cut it in this world. So there's an example where in fact you have no solutions.
How about this, here's one: 7(x + 2) = 3x - 5 + 4x. Well, same thing here, I'm going to try to pull the x's together, so I'll do this in two steps. Step 1 is to distribute on this side, so I'm going to parallel process here, this is what's hot now. So I see 7x, and then don't forget the plus 14, 2 times 7, and that equals - and here I have a 3x plus 4 more x's - total net gain: 7, 7x, and then a -5. I'll move this to the other side, so I'll subtract 7x on both sides, and maybe now you can already see it happening. Here they actually drop out, here I have 14, here these drop out, and I'm left with -5. Whoops, that is not too good; 14 does not equal -5, and that's an equivalent equality to this, which means this is a contradiction. There are no values for x, which will actually satisfy this, so in fact, no solutions.
And one final one to just end it up on a high note maybe, or low note, let's see. Here we go. How about 7(x + 2) = 2x -5 + 4x. It almost looks exactly like the previous one. In fact, the only difference is right here. All right, really fast, here we go. Same process, if I distribute, I see 7x plus 14 equals - I combined 4 and 2 now is 6x -5. Now I'm going to bring that 6x over to the other side, to the Netherlands. I'm going to bring it to the dark side. Now notice that now they don't cancel on this side - 7x - I take away 6x, it leaves me with just 1x plus the 14, and here they cancel them up with -5. So this actually can be solved, and now subtract 14 from both sides - both sides, make sure it balances - and I'm left with x plus zero equals -19. So this just slight alteration from this question, this actually has a solution, while this one is a contradiction, has no solutions. So you have to be very careful and make sure you work it through, make sure whether something has a solution, or whether it doesn't. So watch out for those equations for really there are no solutions. I'll see you after the next lecture.
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Equivalent Equations and Equations with No Solution Page [2 of 2]