Quadratic systems: both variables are squared

What are the solutions to the system of equations y is equal to 1/2 x and 2x squared minus y squared is equal to 7. And they say solutions, because these two, if we view them as two curves, they could very well intersect with each other twice. So let's see what's going on here. We have y is equal to 1/2 x and 2x squared minus y squared is equal to 7. And the best way to approach these is to just try to substitute one constraint into the other constraint, or substitute one equation into the other one. It seems easier to substitute 1/2 x for y into this equation, because they've already solved for y here. Here, it's much harder to solve for x or y, so let's do that. Every place we see a y here, let's substitute it with this, that y must also be equal to 1/2 x, and then see if we can solve for x. So on this equation, we have 2x squared minus y squared. But now we're saying that y must also be equal to 1/2 x. And that is going to be equal to 7. Now let's see if we can solve for x doing a little bit of algebraic manipulation. So we get 2x squared minus. So 1/2 squared is 1/4, and then x squared. So we could say x squared over-- let me write that as 1/4. So let's say it's 1/4 x squared is equal to 7. So I have 2x squareds, and I subtract out a 1/4 x squared, so I'm going to have a 1 and 3/4 x squared. Or you could view this as 8/4 minus 1/4 is 7/4 x squared. 7/4 x squared is equal to 7. Multiply both sides times the reciprocal of 7/4, so 4/7. Multiply both sides by 4/7. And we get x squared is equal to 4. And so x could be positive or negative 2. It's the positive and negative square root of 4. So x is equal to the plus or minus square root of 4. x is equal to positive 2 or negative 2. Now, given that x is positive 2 or negative 2, let's substitute back into either of these equations to figure out what y is, what the corresponding y is for each of these. So if x is 2, y is going to be 1/2 of that. It's going to be 1. So we have the point 2 comma 1. All I did is say, look, x is 2. 1/2 times 2 is 1. If x is negative 2, then y is going to be 1/2 times negative 2, which is going to be negative 1. And both of these definitely satisfy this first constraint. And you can verify that they also satisfy this second constraint right over here. 2 times 2 squared is-- well, 2 squared is 4. 2 times that is 8. Minus 1 squared is 7. 2 times negative 2 squared is still-- this whole thing's going to be 8. Minus negative 1 squared still equals 7. So the solutions to the system of equation-- one is the coordinate 2 comma 1. x is 2. y is 1. The other is the coordinate negative 2, negative 1. x is negative 2, and y is negative 1.