This is a problem about an elliptic quartic in P^{3}

Y is defined by
x^{2} - xz - yw = 0 and
yz - xw -zw = 0
Let P be the point (0,0,0,1)

Let f be the projection from P to the plane w=0.

I need to show that f induces an isomorphism of Y - P with the plane cubic curve K defined by:
zy^{2} - x^{3} + xz^{2}=0
minus the point Q:=(1,0,-1)

The point of the exercise is to prove that Y is irreducible and nonsingular. This follows after showing that f is an isomorphism since then we can show that K is irreducible and nonsingular which is straightforward.

I have been trying to show that f(Y) is contained in K and vice versa.

Am I correct to right the f(Y) as the equations that define Y with w==0?

if so f(Y)= x^{2} - xz = 0 = yz (excluding the point (0,0,0))

But the algebra is not working out for me. Is there another approach?