Suppose $\mathcal{L} = f(x,y) + \lambda g(x,y,z)$ where $z$ is actually constant which I would like to take to use implicit function on.
$$\frac{\partial \mathcal{L}}{\partial x} = f_x(x,y)+\lambda g_x(x,y,z)=0$$
$$\frac{\partial \mathcal{L}}{\partial y} = f_y(x,y)+\lambda g_y(x,y,z)=0$$ After rearrange the euquation, I have $$F(x,y,z) =\frac{f_x(x,y)}{f_y(x,y)} - \frac{ g_x(x,y,z)}{ g_y(x,y,z)} = 0 $$ Can I apply implicit function on $F(x,y,z)$ to find $\frac{dx}{dz}$?