/sci/ - Science & Math

How do i find the points of intersection of the curves [math] x=1 [/math] and [math] y=x^2 [/math] ?

To start, I put them in projective space with homogenization parameter z and set them equal:

[eqn] x-z=yz-x^2 [/eqn] and then got [math] x(1+x)=z(1+y) [/math] .

Now clearly (x,y,z)=(1,1,1) is a point of intersection and corresponds to the real point (1,1). But it also seems to me that (-1,-1,z) works and so does (0,-1,z) and (0,y,0) among others. But there can only be one more point of intersection and i know its at the point at infinity which corresponds to z=0. So are all those points i just said the same or what? Basically what do the points (-1,-1,z) and (0,-1,z) represent when z is not 0? Only thing i can think is that they are nonsense since the first implies -1=1 and the second that 0=1 and 0=-1. But arent they technically solutions to the simultaneous equations?