Hey, everyone. Welcome back. So earlier in this course, we learned how to plot and solve a single equation on a graph, something like y=8x-4. But now you might start to see problems that have multiple equations in them. You'll be asked to solve these types of problems, and this is what your book will refer to as a system of equations. Now in this video, we're only going to focus on linear equations. But, basically, whenever you see a system of equations, it really just means that there are multiple equations like we have in our problem. So I'm going to explain the basic difference between these types of problems, and then later on, we'll learn a bunch of different ways to solve these types of problems. Let's go ahead and get started. So remember that the solution to a single equation is you're really just trying to find x y pairs that satisfy this equation. So for example, if we were asked to determine if each point was a solution to this equation, these three points over here, all we'd have to do is we could plug them into our equation and figure out if they make true statements, but we can also just graph them and see if they're on the line. Graphically just means if these points are solutions, that means that they're on the line. So for example, something like (-2,0). So if you plot that point, that's going to be over here. Is this a solution to this equation? Well, it's not on the line, so it's not. If you were to plug this into this equation, it would lead to a false statement. How about this point, (0,-4)? Well, that's going to be over here. It's on the line, so, therefore, this is a solution to the system of, or sorry, to this specific equation. If you plug these numbers into this equation, you'll get a true statement. Now what about the point (1,4)? If you plot that out, what you'll see is that this is right over here. So is this? Well, this actually is on the line, so it also is a solution to this equation. So, it's pretty straightforward. If it's on the line, it satisfies this equation.

Now let's take a look at this situation over here where we have multiple equations, and I've already plotted them for you. If you plot them in slope-intercept form, you'll see that the graph indeed actually does look like this. When we're asked to figure out if a point is a solution to the system of equations, really, what you're doing here is to solve these types of problems, you're going to find coordinate pairs or x y pairs that satisfy not just one equation, but all of the equations. So, for example, if you were to look at these points over here, (0,-4), what we'll see here is that this only is on the blue line, but this is not a solution to the system of equations because it's not on the red line. It doesn't satisfy both of them at the same time. So this is not a point; this is not a solution to the system because this only satisfies the blue equation only. Let's take a look at the next one. The next one over here is the point (2,5). If you plot that out, that's going to be right over here. So is this a solution to the system of equations? Well, this is kind of like the opposite of a, where it only satisfies the red equation. Right? So it's on the red line, but notice how it's not on the blue line. So this satisfies the red only, but not the blue, and so that's why it's not a solution to both. Let's take a look at our last point, which is (1,4). So the point 1 comma 4, you'll see, is actually right over here. And if you take a look at this, is this a solution to the system? It satisfies the red line, but it also satisfies the blue line. So, it turns out that this is a solution to the system of equations because it satisfies both the lines. It's on both of the lines. Alright? So what we can see here is that graphically, the solution to the system of equations is actually just the place where the lines intersect or where they meet. Alright? So you could plug in these points into both of these equations, and you'll get true statements for both of them. But graphically, the solution is actually just the point where the lines cross. Alright? And that's really all there is to it.

So I want to actually point out the big difference between these types of problems because what we saw is that for a single equation, the solution is that you actually had many points that satisfy just one line. So, for example, we could pick these two points, but, really, any point along this line will satisfy just this one equation. Whereas over here, what we saw is that there's only really just one solution that satisfies all the lines. There are lots of points that satisfy one or the other, but there's only one that satisfies all of them. Alright? And that's really the main difference between a single equation and a system of equations. Now one of the things I want to mention is that this is true for most problems where you'll have just one solution. But there are other types of solutions, and we'll deal with them as we sort of go along. Anyway, so that's an introduction to a system of equations. Hopefully, that made sense. Thanks for watching.