Reasoning with linear equations | Solving equations & inequalities | Algebra 1 | Khan Academy

In many other videos, we've taken equations like this and tried to quote solve for x. What we're going to 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 equivalent 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 equivalent statement is I could distribute this 3 onto the x and/or onto the x plus 1, and then this part of it could be rewritten as 3x plus 3, and then we have minus x is equal to 9.

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 a given x, the other one will be true for that same x as well, and vice versa.

We can write other equivalent statements; for example, if we were to combine x terms here, if we were to take 3x and then minus x right over there, I could rewrite that as 2x, and then I have plus 3 is equal to 9. Now all three of these statements are equivalent. If there's an x where 2 times that x plus 3 is equal to 9, then it is also the case that 3 times that x plus 1 minus x is equal to 9, 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. We can do other equivalence preserving operations, and you've seen them before. You could subtract 3 from both sides. In general, if you are adding or subtracting the same value from both sides, it is equivalence preserving.

If you are 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 3 from both sides, and you would get 2x is equal to 6.

Once again, any x that satisfies this last equation will satisfy any of the other equations, and vice versa. Anything that satisfies any of these other equations will satisfy this last one, so they are all equivalent to each other.

Another equivalence preserving operation is to multiply or divide both sides by a non-zero constant. Here, we could divide both sides by 2; 2 is not 0, and it's constant. If we did that, we will get another equivalent statement that x is equal to 3.

So any x that satisfies this—and there's one, x equals 3—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.

One way to think about it: adding the same number to both sides of an equation is equivalence preserving. Multiplying or dividing both sides by a non-zero constant value is equivalence preserving. Distributing, like we did in this first step, is equivalence preserving. Combining like terms is equivalence preserving.

Now you're probably saying, "Well, what are some non-equivalence preserving operations?" Well, imagine something like this. Let me just start with something very obvious. If I said that x is equal to 2, a non-equivalence preserving operation is if I were to add, 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 2, and it is not the case that anything that satisfies this second equation satisfies the top equation, or vice versa. x equals 2 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 3, I would get x is equal to 6. Well, by only multiplying the right-hand side by a value, it's not the case that anything that satisfies x equals 6 will satisfy x equals 2. That is somewhat obvious here.

Now there are a little bit trickier scenarios. Let's say we have the equation 5x is equal to 6x. Now one temptation is, "Well, I want to do the same thing to both sides." I could just divide both sides by x. What will happen in that scenario?

Well, if you divide both sides by x, you could think that an equivalent statement is that 5 is equal to 6, and you know that there's no x for which 5 is equal to 6. You can't make 5 equal to 6 or 6 equal to 5. And so this would somehow make you imply that, okay, there's no x that can satisfy.

If you assume that these were equivalent statements, you'd say there's nothing that could satisfy 5 equals 6. 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.

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 5x from both sides.

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 0 is equal to x. Now 0 equals x and 5x equals 6x, these are equivalent 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 a non-zero value, and that's going to be equivalence preserving.

Hopefully, you 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 let me start over here—if I had 2x is equal to 6, and if I were to multiply both sides by zero, I would get 0 is equal to 0.

And 0 equals 0 is true for any x. Zero is always going to be equal to zero, but the problem is that first statement isn't true for all x; it's only true for x equals 3. So these two are not equivalent statements; they have a different set of x's that will satisfy them.

You have to be very careful when you're dealing with things that are either r or zero. You can add or subtract zeros—obviously, that's not going to change things much. When you multiply both sides by zero, you can start getting things that are not equivalent statements. When you multiply or divide by things that could be zero, like variables, that also is a dangerous game to play.