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Reasoning with linear equations

When we perform operations to manipulate equations, some operations produce equivalent equations, while others don't necessarily produce equivalent equations. When we're solving an equation, we need to use operations that guarantee equivalence. Created by Sal Khan.

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Video transcript

- [Instructor] In many other videos, we've taken equations like this and tried to quote solve for X. What we're gonna do in this video is deepen our understanding a little bit about what's going on and really think about the notion of equivalence or equivalence statements. So let me write that down, equivalence. Now, what do I mean by equivalence? Well, I'll use this equation here to essentially keep rewriting it in equivalent ways and I'll talk a little bit more about what that means. So one thing that I could do to essentially write the same equivalence statement is I could distribute this three onto the X or onto the X plus one and then this part of it could be rewritten as three X plus three and then we have minus X is equal to nine. Now, what I will say or what it might be obvious to you on some level is this top equation and this second equation are equivalent. What does that mean? It means if one of them is true for given X, the other one will be true for that same X as well, and vice versa and we can write other equivalence statements. For example, if we were to combine X terms, if we were to take three X and then minus X, right over there, I could rewrite that as two X and then I have plus three is equal to nine. Now all three of these statements are equivalent. If there's an X where two times that X plus three is equal to nine, then it is also the case that three times that X plus one minus X is equal to nine and vice versa. If there's some X that would make this top equation true then it's going to make this last equation true. And we can do other equivalence preserving operations and you've seen them before. You could subtract three from both sides. In general, if you are adding or subtracting the same value from both sides, it is equivalence preserving. If you're distributing a value like we did in that first step that is equivalence preserving. If you combine like terms so to speak, that is equivalence preserving and so here we'll do an equivalence preserving operation. We'll subtract three from both sides and you would get two X is equal to six and once again, any X that satisfies this last equation will satisfy any of the other equations and vice versa. Any X that's satisfies any of these other equations will satisfy this last one and so they are all equivalent to each other. And then another equivalence preserving operation is to multiply or divide both sides by a non zero constant. And so here we could divide both sides by two, two is not zero and it's constant and if we did that, we will get another equivalent statement that X is equal to three. So any X that satisfies this and there's one, X equals three would satisfy the other ones and any X that satisfies any of the other ones would satisfy this last one so these are all equivalent. So one way to think about it, adding the same number to both sides of an equation that is equivalence preserving. Multiply or dividing both sides by a non-zero constant value is equivalence preserving. Distributing like we did in this first step equivalence preserving, combining like terms equivalence preserving. Now you're probably saying, well, what are some non equivalence preserving operations? Well, imagine something like this, lemme just start with something very obvious. If I said that X is equal to two a non equivalence preserving operation is if I were to add or subtract or multiply or divide only one side of this equation by a value let's say I only added one to the left side. Then I would have X plus one is equal to two and it is not the case that anything that satisfies this second equation satisfies the top equation or vice versa. X equals two clearly satisfies the top equation but it doesn't satisfy the second one, that's because we did a non equivalence preserving operation. Likewise, if I only multiplied the right hand side by three, I would get X is equal to six. Well, by only multiplying the right hand side by value it's not the case that anything that satisfies X equals six will satisfy X equals two, that is somewhat obvious here. Now there's a little bit trickier scenarios, let's say we have the equation five X is equal to six X. Now, one temptation is, well, I wanna do the same thing to both sides, I could just divide both sides by X, what'll happen in that scenario? Well, if you divide both sides by X you could think that an equivalence statement is that five is equal to six and you know that there's no X for which five is equal to six. You can't make five equal six or six equal five and so this would somehow make you imply that okay, there's no X that can satisfy. If you assume that these were equivalence statements, you'd say there's nothing that could satisfy five equal six so maybe there's nothing that satisfies this top equation. But this actually isn't an equivalence preserving operation because you're actually dealing with a scenario where X is equal to zero and you're dividing by zero. And so you have to be very careful when you're dividing by a variable, especially if the variable that makes that equation true happens to be zero. And so in order to be clear that you're preserving equivalence here, the way that I would tackle it is I would subtract five X from both sides and if you do that, and that is an equivalence preserving operation, you could subtract that expression from both sides or that term from both sides and then you would be left with zero is equal to X and now zero equals X and five X equals six X, these are equivalence statements. They are equivalent equations anything that makes this one true is going to make that one true and anyone that makes that one true is going to make this one true. Now one last thing you might have heard me say, you can multiply or divide by non-zero value and that's going to be equivalence preserving. And hopefully you've just got a sense of why dividing by zero is not a good idea. In fact, dividing by zero is always going to be a strange thing and it's undefined, but also multiplying by zero. For example, if I had, let's say, actually lemme start over here. If I had two X is equal to six, and if I were to multiply both sides by zero, I would get zero is equal to zero and zero equals zero is true for any X. Zero is always going to be equal to zero but the problem is, is that first statement isn't true for all X it's only true for X equals three. So these two are not equivalent statements. They have a different set of Xs that will satisfy them and so you have to be very careful when you're dealing with things that are either are zero you can add or subtract zeros, obviously that's not gonna change things much, but when you multiply both sides by zero, you can start getting things that are not equivalence statements and when you multiply or divide by things that could be zero, like variables, that also is a dangerous game to play.