Hey, everyone. So up until now, when we've solved equations, they've always been of 1 variable, like x+2=5. But equations in this course won't always be so simple. Instead of just one variable, a lot of equations in this course will now start to involve 2 variables, and the most common ones you'll see are going to be x and y. So for example, instead of x+2=5, now we're going to have x+y=5. I'm going to show you the main difference between these two types of equations and how to solve them and, more importantly, how to visualize them. So let's go ahead and get started here.

So with equations with one variable, like, for example, x+2=5, you're really just trying to find the value of x that makes this equation true. So you would isolate and solve for x. You would subtract from both sides, and your answer ends up being 3. That's the number that makes this equation true. We visualize it by plotting it on a one-dimensional number line, like this point over here. Now let's take a look at x+y=5. So now we actually have 2 variables, x and y. Both are letters that can be replaced with numbers. So how do we solve this? Well, one thing you might notice is that if you try to isolate one of the variables, it's not really going to be much help because x=5-y doesn't give you any information about what x or y could be. Let's think about what this equation means. Can I think of 2 numbers that, when I add them together, I get 5? And, actually, yes, I can. Because, for example, if x=1, what do I have to add to 1 to get to 5? Well, y could be 4. So that's a solution to this equation. Both of these numbers here, this combination makes this equation true. But is this the only combination of numbers that makes this true? Well, actually, no. Because what if x=2? If x=2, then y could be 3, and that also satisfies this equation, x+y=5. In fact, you could have another one, x=5, and then y could just equal 0. So the whole point here is that with equations with 1 variable, the solution was always just one number. It was a single point on a one-dimensional plane versus when you have equations with 2 variables, as we can see, you end up with many solutions that satisfy this equation. And the way that we represent them is not on a number line, but actually as points as ordered pairs x,y on a two-dimensional plane.

So, basically, what happens is I can take this x=1,y=4 and turn it into an ordered pair 1 comma 4. This becomes 2 comma 3, and this becomes 5 comma 0. And we can plot these on a 2-dimensional plane. So, for example, 1 comma 4, this point satisfies this equation. 2 comma 3, that also satisfies this equation. And 5 comma 0 also satisfies this equation. So all these things actually satisfy. In fact, there's actually an infinite number of solutions because if you sort of keep this pattern going on here, any points that are basically on this little line over here will actually satisfy between these two types of equations.

Now how do we actually use this in problems? Well, let's take a look at our example here. In our example, we have this equation, x+y=5. This is the same exact equation we've been using before, and we want to first determine if these points satisfy the equation. We've got 3 comma 2, 4 comma 1, 0 comma 0, and negative 1 comma 6. So in order if you're if you're ever asked to determine whether points x,y points satisfy an equation, what they're really just asking you to do is they're asking you to replace the x and y values to check if the equation is true. Remember, that's what satisfying an equation means. So what we're doing in part a is we're when we have a coordinate like 3 comma 2, this is really just giving us an x and a y value, and we just plug it into this equation, x+y=5, and we just figure out if that makes the equation true. So does it? Well, if I basically just place x with 3 and y with 2, then 3 plus 2 does, in fact, give me 5. So this definitely does satisfy the equation. Let's get started with the next one, which is 4 comma 1. Again, does this satisfy x+y=5? Well, if I replace x with 4 and y with 1, this does indeed get me 5. So both of these points do satisfy the equation. What about 0 comma 0? I do 0 comma 0 into x+y=5, then I'm just replacing x with 0, y with 0, and 0 plus 0 does not give me 5. So this actually does not turn out to be a solution to this equation. And last but not least, what about negative 1 comma 6? So into x+y=5. Well, this is going to be negative 1 plus 6, and this also does not give me 5. It gives me 5. So it turns out that not all of these points are going to satisfy the equation. The first two work. The second two didn't work. And now we're going to go ahead and plot them. Right? So as we can see the point 3 comma 2 is going to be here. That's the point 3 comma 2. Right? That's this point over here. We've got 4 comma 1. This is also going to be a solution to the equation. And then we've got 0 comma 0, which is over here, and then negative 1 comma 6. So if you've seen here, there's a pattern that happens. Basically, when points do satisfy an equation, when they do make the equation true, then they are on the graph of that equation versus when they do not satisfy an equation, as we've seen with these points over here, they are not on the graph of that equation. So that's the basic difference between equations with 2 variables versus 1 variable. Alright? So, hopefully, that made sense. Thanks for watching.