group of points which this eqution determines

Question

I suspect that $xy+x=0$ determines the equation of an infinite line in $R^3$ for which a general coordinate is $(0,-1,z)$ since in the end, it is the intersection of x=0 and y=-1 (the planes). is this right?

Answer 1

The equation is independent in $z$, so any $z$ coordinate will do. Factoring, we have $x(y+1)=0$, so we must have $x=0$ or $y=-1$. If $x=0$, then both $y$ and $z$ are independent, so we have the plane $x=0$, i.e. the y-z plane. Similarly, if $y=-1$, we have $x,z$ are independant, so we have a plane in the x-z axis going through $y=-1$. So the solution is the union of two planes