Answer
See the explanation below.
Work Step by Step
When $F$ is differentiable, then $\dfrac{\partial z}{\partial x},\dfrac{\partial z}{\partial y}$ can be calculated as:
$\dfrac{\partial z}{\partial x}=-\dfrac{\dfrac{\partial F}{\partial x}}{\dfrac{\partial F}{\partial z}}$
and
$\dfrac{\partial z}{\partial y}=-\dfrac{\dfrac{\partial F}{\partial y}}{\dfrac{\partial F}{\partial z}}$
