Please consider $f(x,y,z)=0$, such that $f$ is a $C^{\infty}$ function in the neighborhood of $p=(p_1,p_2,p_3)$ on $\mathbb{R}^3$, and $f(p)=0$. Also assume that $$ \dfrac{\partial f}{\partial x}(p) \neq 0,$$ $$ \dfrac{\partial f}{\partial y}(p) \neq 0,$$ $$ \dfrac{\partial f}{\partial z}(p) \neq 0.$$
Therefore, implicit function Theorem, implies that each variable can be solved locally, in terms of the other variables. So, for example, we have $z=z(x,y)$ in a suitable neighborhood of $(p_1,p_2)$.
Now assume that furthermore we have \begin{equation} \dfrac{\partial^2 z}{\partial x^2} \dfrac{\partial^2 z}{\partial y^2}-\left(\dfrac{\partial^2 z}{\partial x \partial y}\right)^2=0. \end{equation}
Then, show that for the functions $x=x(y,z)$ and $y=y(x,z)$, the similar relations like above holds, i.e.
\begin{equation} \dfrac{\partial^2 x}{\partial y^2} \dfrac{\partial^2 x}{\partial z^2}-\left(\dfrac{\partial^2 x}{\partial y \partial z}\right)^2=0. \end{equation}
\begin{equation} \dfrac{\partial^2 y}{\partial x^2} \dfrac{\partial^2 y}{\partial z^2}-\left(\dfrac{\partial^2 y}{\partial x \partial z}\right)^2=0. \end{equation}
My idea: I think it is nothing except calculation. By calculating $z_{xy}$, $z_{xx}$, and $z_{yy}$, and then putting them in the given relation we find some relation between the partial derivatives of $f$. Then using this information on $f$, we may continue calculation to obtain the desired result.
Am I thinking in the right way? Or this problem can be solved more easily (not tedious calculations)?
Thanks.