Twisted cubic as a complete intersection

Question

Let $C$ be the twisted cubic in $\mathbb{P}^3$ cut out by $xz-y^2, yw-z^2, w(xw-yz)$. I found online that this is a set-theoretic intersection of the quadric surface $xz-y^2 = 0$ and the cubic surface $z(yw-z^2)-w(xw-yz)=0$. I am able to check that these indeed set-theoretically cut out the twisted cubic, but where does this come from?