You have:

F(x,y,z,w)=0

G(x,y,z,w)=0

So

F_{x}dx+F_{y}dy+F_{z}dz+F_{w}dw=0

G_{x}dx+G_{y}dy+G_{z}dz+G_{w}dw=0

Solve for dz and dw:

dz=Adx+Bdy

dw=Cdx+Ddy

A=dz/dx, B=dz/dy, C=dw/dx, D=dw/dy

That’s the principle. The rest is doing the partial derivatives F_{x}..., and algebra, if that interests you.

EDIT: Or solve for dy and dw as functions of dx and dz to get partial you want.