 In this video I want to solve a nonlinear system of equations, but I want to show you what it looks like if you have an inconsistent system So notice here. We have the equation x squared minus y squared equals 4, which is actually a hyperbola That you see here on the screen and then also the equation Michael's x squared. That's your standard parabola Which you can see right here So if you graph these things to scale we can very well see that there is no point on the green grab parabola and the yellow Hyperbola, so this should be an example of an inconsistent system We can see that geometrically very simply But how would you detect that algebraically as we try to solve systems of linear equations earlier in this lecture series? We often we get things like 0 equals 7 which of course is a contradiction which would indicate you have an inconsistent system That thing is still present, right? If you get an ink if you get a contradiction that means you're not gonna have a solution in that regard, right? But another thing that can show up with these nonlinear systems It could be that there's no contradictions that emerge But it could it could the situation could be that there's no real solutions If you have to solve an equation that has no real solutions that also would indicate inconsistency So let me show you how we could do that one here and I'm going to solve this you could do it by Substitution you could just substitute y equals x squared right here That would give you the equation x squared minus x to the fourth equals four This would be the substitution approach and that does require we solve a degree four polynomial It's by no means I mean it's something we definitely could do. It's not horrible But I actually think elimination works out a little bit better on this one because if we take the hyperbola x squared minus y squared equals four and we compare that with the Parabola, which I'm going to times the parabola by negative one Let's do a little bit of a massage into it first Let's move the y to the other side so we get x squared minus y equals zero and then times that by negative one Negative one on both sides. You're going to get negative x squared plus y Equals zero let's add those together So that the x squareds cancel And we end up with a negative y squared plus y equals four Let's just move these friends to the other side of the equation You get zero equals y squared minus y plus four and so as you try to factor this You're like okay factors of two that have to be negative one You'd have to take negative one negative four, which is negative five doesn't work negative two negative two is negative four That doesn't work. Okay. I'll use the quadratic formula So we kind of sing pop goes the weasel on our head here y equals negative b Plus or minus the square root now you go through the song there You're gonna get a plus one plus or minus the square root of negative one squared minus four times One times four all over two right Simplifying this thing here You're gonna get one plus or minus the square root of negative 15 over two and so this is this is the This is the point right here notice our discriminant turned out to be negative 15 That's less than zero that tells us there are no real solutions. So there are no real numbers There are no real numbers that will make this intersection happen if I want to graph this thing with complex variables Then we'll get intersections But for the real number system these things don't intersect and that's what we mean here by inconsistent because what does inconsistent mean? Inconsistent means there's no solution and in this context it means there's no real solution So if you have to solve quadratic equations with negative discriminants That's gonna indicate to you that you have no real intersections between your curves