How does I solve?

If one has the problem x=yx (or P=NP), the answer could either be y=1, or x=0.
y=1 can be obtained by:
x=yx
x/x=yx/x
1=y

But, how does one obtain x=0?

For y=1 to be true, x cannot be 0.  You have to restrict your equation.