Reasoning with systems of equations | Systems of equations | Algebra I | Khan Academy

In a previous video, we talked about the notion of equivalence with equations. Equivalence is just this notion that there's different ways of writing what are equivalent statements in algebra. I could give some simple examples. I could say 2x equals 10 or I could say x equals 5. These are equivalent equations. Why are they? Because an x satisfies one of them if and only if it satisfies the other. You can verify that in both cases, x equals 5 is the only x that satisfies both.

Another set of equivalent equations you could have is 2x is equal to 8 and x equals 4. These two are equivalent equations, and 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. 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. Let's say this system tells us that there's some xy pair where 2 times that x plus that y is equal to 8, and that x plus that y is equal to 5. Now, we can have an equivalent system if we replace either of these equations with an equivalent version. For example, many of you, when you look to try to solve this, might say, well, if this was a negative 2x here, maybe I could eventually add the left side.

We'll talk about why that is an equivalence preserving operation. But in order to get a negative 2 out here, you'd have to multiply this entire equation times negative 2. And so if you did that, if you multiplied both sides of this times negative 2, what you're going to get is negative 2x minus 2y is equal to negative 10. This equation and this equation are equivalent. Why? Because any xy pair that satisfies one of them will satisfy the other, or an xy pair satisfies one if and only if it satisfies the other.

If I now think about 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 any xy pair, if an xy 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. We've done this before, and 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.

One way to think about it is what I'm going to do to create an equivalent system here. I am going to keep my first equation, 2x plus y is equal to 8, but then I'm going to 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 the equation, you get an equivalent equation. So I'm going to do that over here, but it's going to be a little bit interesting.

So if you had negative 2x minus 2y is equal to negative 10, and what I want to do is I want to add 8 to both sides. So I could do it like this; I could add 8 to both sides. But remember our system is saying that both of these statements are true, that 2x plus y is equal to 8 and negative 2x minus 2y is equal to negative 10.

So instead of adding explicitly 8 to both sides, I could add something that's equivalent to 8 to both sides. I know something that is equivalent to 8 based on this first equation. I could add 8, and I could do 8 on the left-hand side, or I could just add 2x plus y.

Now, I really want you—you might want to pause your video—say, okay, how can I do this? Why is Sel 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 xy pair satisfies one equation if and only if it satisfies the other. Here, we know that 2x plus y needs to be equal to 8.

So if I'm adding 2x plus y to the left and I'm adding 8 to the right, I'm really just adding 8 to both sides, which is equivalence preserving. When you do that, you get these negative 2x and 2x cancel out, and you get negative y is equal to negative 2. So I can rewrite that second equation as negative y is equal to negative 2.

I know what you're thinking. You're like, wait, when 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. 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, both equations are there the whole time; you're just rewriting them in equivalent ways.

Once again, this system, this system, and this system are all equivalent. Any xy pair that satisfies one will satisfy all of them and vice versa. Again, we can continue to rewrite this in equivalent ways. That second equation, I could multiply both sides by negative one—that's equivalence preserving. If I did that, then I get—I haven't changed my top equation, 2x plus y is equal to 8.

On the second one, if I multiply both sides by negative 1, I get y is equal to 2. Once again, these are all equivalent systems. I know 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 xy pair is. If we know that y is equal to 2, and we know that that's true in both equations—remember, it is an "and" here—we're assuming there's some xy pair that satisfies both 2x plus y needs to be equal to 8 and y is equal to 2.

That means up here where we see a y, we can write an equivalent system where instead of writing a y there, we can write a 2 because we know that y is equal to 2. We can rewrite that top equation by substituting a 2 for y. So we could rewrite that as 2x plus 2 is equal to 8, and y is equal to 2. This is an "and" right over there; it's implicitly there.

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 2 from both sides of that top equation? It's still going to be an equivalent equation.

So I could rewrite it as if I subtract 2 from both sides, I'm going to get 2x is equal to 6, and then that second equation hasn't changed: y is equal to 2. There's some xy 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.

Then, of course, this top equation, an equivalence preserving operation is to divide both sides by the same non-zero value. In this case, I could divide both sides by 2, and then I would get—if I divide the top by 2, I would get x equals 3 and y is equal to 2. 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 xy pair actually is. In the past, you might have just, you know, assumed that you can add both sides of an equation or do this type of elimination or do some substitution to just quote figure out the x and y's. 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 xy pair is that satisfies both equations in the system.