Answer
$(\dfrac{\partial z }{\partial x})(\dfrac{\partial x }{\partial y})(\dfrac{\partial y}{\partial z})=-1$
Work Step by Step
Use equation 7: $z_x=-\dfrac{F_x}{F_z}$ and $z_y=-\dfrac{F_y}{F_z}$
$(\dfrac{\partial z }{\partial x})(\dfrac{\partial x }{\partial y})(\dfrac{\partial y}{\partial z})=- \dfrac{F_x}{F_y}\dfrac{F_y}{F_x}\dfrac{F_z}{F_y}$
Since, $z_x=-\dfrac{F_x}{F_z}$ and $z_y=-\dfrac{F_y}{F_z}$
Re-write as: $(z_x)(x_y)(y_z)=- \dfrac{F_x}{F_y}\dfrac{F_y}{F_x}\dfrac{F_z}{F_y}$
This gives:
$(\dfrac{\partial z }{\partial x})(\dfrac{\partial x }{\partial y})(\dfrac{\partial y}{\partial z})=-1$
