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Nonlinear equation graphs — Basic example

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- [Instructor] A system of equations and their graphs in the xy-plane are shown above. How many solutions does the system have? And I put the picture here on the side. So we have three equations here that represent relations between x and y. And so we have y squared is equal to x plus three. So we could even show which ones are which. This one is this one right over here. That's this sideways parabola that we're seeing right over, actually, I'm having trouble tracing it. So we have this one right over there. We have x, or y plus x is equal to negative one. That's just going to be this line right over here. That is a linear equation. So that's just this line right over here. And the last one, the last one, x squared plus y squared is equal to five, that's equal to that circle. Now a solution for the system, the system that has three equations, two of which are nonlinear, in order to be a solution to this system, it needs to be a point that is common to all three curves or all three graphs or a point that's common to all or a point that satisfies all three of these. So if we see, so you might be tempted to say, okay, what about this point right over here? But this point satisfies the circle, and it satisfies the sideway parabola. But this point doesn't lie in the line. In order to satisfy them, you have to, and this one over here, likewise, it satisfies the sideways parabola and the circle, but it does not satisfy the line. In order to be a solution for the system, it has to be an xy-coordinate that satisfies all of these three. And the ones that satisfy all three, well, this point right over here, all three lines intersect at exactly that point, or at least it looks like they do. And all three points intersect at exactly that point right over here. So this solution has two, or this system has two solutions. The four is tricky 'cause right when your eyes see this, you see four points of intersection very naturally, but you need, it's very important to realize that this point is only, has two of curves intersecting, not, and this one is well, only two of the curves intersecting, not all three.