Reasoning with systems of equations

- [Sal] In a previous video, we talk about the notion of equivalence with equations. And equivalence is just this notion that there's different ways of writing what our equivalent statements in algebra. And I can give some simple examples. I could say two x equals 10, or I could say x equals five. These are equivalent equations. Why are they? Because an x satisfies one of them if and only if it satisfies the other. And you can verify that in both cases, x equals five is the only x that satisfies both. Another set of equivalent equations, you could have two x is equal to eight and x equals four. These two are equivalent equations. An x satisfies one if and only if it satisfies the other. In this video, we're going to extend our knowledge of equivalence to thinking about equivalent systems. And really, in your past when you were solving systems of equations, you were doing operations assuming equivalence, but you might not have just been thinking about it that way. So let's give ourselves a system. So let's say this system tells us that there's some x y pair where two times that x plus that y is equal to eight, and that x plus that y is equal to five. Now we can have an equivalent system if we replace either of these equations with an equivalent version. So for example, many of you when you look to try to solve this, you might say well, if this was a negative two x here, maybe I could eventually add the left side. And we'll talk about why that is an equivalence preserving operation. But in order to get a negative two out here, you'd have to multiply this entire equation times negative two. And so if you did that, if you multiplied both sides of this times negative two, times negative two, what you're going to get is negative two x minus two y is equal to negative 10. This equation and this equation are equivalent. Why? Because any x y pair that satisfies one of them will satisfy the other, or an x y pair satisfies one if and only if it satisfies the other. And so if I now think about the system, the system where I've rewritten this second equation and my first equation is the same, this is an equivalent system to our first system. So these, any x y pair, if an x y pair satisfies one of these systems, it's going to satisfy the other and vice versa. Now the next interesting thing that you might realize, and if you were just trying to solve this, and this isn't an introductory video in solving systems, so I'm assuming some familiarity with it, you've probably seen solving by elimination where you say okay, look, if I can somehow add these, the left side to the left side and the right side to the right side, these x's will quote cancel out and then I'll just be left with y's. And we've done this before. You can kind of think you're trying to solve for y. But in this video, I want to think about why you end up with an equivalent system if you were to do that. And one way to think about it is what I'm going to do to create an equivalent system here is I am going to keep my first equation, two x plus y is equal to eight. But then I'm gonna take my second equation and add the same thing to both sides. We know if you add or subtract the same thing to both sides of an equation, you get an equivalent equation. So I'm gonna do that over here. But it's gonna be a little bit interesting. So if you had negative two x minus two y is equal to negative 10, and what I want to do is I want to add eight to both sides. So I could do it like this. I could add eight to both sides. But remember, our system is saying that both of these statements are true, that two x plus y is equal to eight and negative two x minus two y is equal to negative 10. So instead of adding explicitly eight to both sides, I could add something that's equivalent to eight to both sides. And I know something that is equivalent to eight based on this first equation. I could add eight, and I could do eight on the left hand side, or I could just add two x plus y. So two x plus y. Now I really want you, you might want to pause your video and say okay, how can I do this? Why is Sal saying that I'm adding the same thing to both sides? Because remember, when we're taking a system, we're assuming that both of these need to be true. An x y pair satisfies one equation if and only if, only if, if and only if it satisfies the other. So here, we know that x, two x plus y needs to be equal to eight. So if I'm adding two x plus y to the left and I'm adding eight to the right, I'm really just adding eight to both sides, which is equivalence preserving. And when you do that, you get, these negative two x and two x cancels out, you get negative y is equal to negative two. And so I can rewrite that second equation as negative y is equal to negative two. And I know what you're thinking. You're like wait, but I'm used to solving systems of equations. I'm used to just adding these two together and then I just have this one equation. And really, that's not super mathematically rigorous because the other equation is still there. It's still a constraint. Oftentimes, you solve for one and then you quote substitute back in. But really, the both equations are there the whole time. You're just rewriting them in equivalent ways. So once again, this system, this system, and this system are all equivalent. Any x y pair that satisfies one will satisfy all of them, and vice versa. And once again, we can continue to rewrite this in equivalent ways. That second equation, I can multiply both sides by negative one. That's equivalence preserving. And if I did that, then I get, I haven't changed my top equation, two x plus y is equal to eight. And on the second one, if I multiply both sides by negative one, I get y is equal to two. Once again, these are all equivalent systems. I know I'm, I sound very repetitive in this. But now, I can do another thing to make this, to keep the equivalence but get a clearer idea of what that x y pair is. If we know that y is equal to two and we know that that's true in both equations, remember, it is an and here. We're assuming there's x y, some x y pair that satisfies both. Two x plus y needs to be equal to eight and y is equal to two. Well that means up here where we see a y, we can write an equivalent system where instead of writing a y there, we could write a two because we know that y is equal to two. And so we can rewrite that top equation by substituting a two for y. So we could rewrite that as two x plus two is equal to eight and y is equal to two. So this is an and right over there. It's implicitly there. And of course, we can keep going from there. I'll scroll down a little bit. I could write another equivalent system to this by doing equivalence preserving operations on that top equation. What if I subtracted two from both sides of that top equation? It's still going to be an equivalent equation. And so I could rewrite it as, if I subtract two from both sides, I'm gonna get two x is equal to six. And then that second equation hasn't changed. Y is equal to two. So there's some x y pair that if it satisfies one, it satisfies the other, and vice versa. This system is equivalent to every system that I've written so far in this chain of operations, so to speak, and then of course, this top equation, an equivalence preserving operation is to divide both sides by a non, the same nonzero value. And in this case, I could divide both sides by two. And then I would get, if I divide the top by two, I would get x equals three, and y is equal to two. And once again, this is a different way of thinking about it. All I'm doing is rewriting the same system in an equivalent way that just gets us a little bit clearer as to what that x y pair actually is. In the past, you might've just, you know, just assumed that you can add both sides of an equation or do this type of elimination or do some type of substitution to just quote figure out the x and y. But really, you're rewriting the system. You're rewriting the constraints of the system in equivalent ways to make it more explicit what that x y pair is that satisfies both equations in the system.