Theorem implicit function

I have the following excercise:

$f,g: \mathbb R^3 \to \mathbb R$ with:

$f(x,y,z)=z^2-2y-xz$

$g(x,y,z)=zy+x^2$

a) Show that in an open environment of $(1,1,-1)$ there exist functions $h_1(z)$ and $h_2(z)$ with $f(h_1(z),h_2(z),z)=0$ and $g(h_1(z),h_2(z),z)=0$

b) Calculate $h_1(z)$ and $h_2(z)$

My ideas for a) :

I define a function $F(x,y,z):=\left(\begin{eqnarray} f(x,y,z)\\ g(x,y,z) \end{eqnarray}\right) $

$F(1,1,-1)=\left(\begin{eqnarray} 0\\ 0 \end{eqnarray}\right) $ so the first requirement for the theorem of implicit function is met.

The Jordan-matrix:$J_F(x,y,z):=\left(\begin{eqnarray} -z & -2 &-x+2z \\ 2x & z & y \end{eqnarray}\right) $

$J_F(1,1,-1):=\left(\begin{eqnarray} 1 & -2 & -3 \\ 2 & -1 & 1 \end{eqnarray}\right) $

$\det(\frac{\partial F}{\partial (x,y)} (1,1,-1))=\det(\left(\begin{eqnarray} 1 & -2 \\ 2 & -1 \end{eqnarray}\right))=3$

so the functions should exist...

Are my ideas correct till here?