I really want to understand this theorem since it's constantly used in economics for simple equations, which I run into often.
Essentially, we have $\frac{\partial F(x,y)}{\partial x} = z$. What conditions do we need for $x = G(y,z)$ for some function $G$, and what are the partial derivatives of $G$ with respect to $y$ and $z$, as functions of $G$ and $F$?
I understand calculus, but not so much about multidimensional jacobian matrices and such, which is what the wikipedia article on implicit function theorem is centered on.
Suppose that $\frac{\partial F (x_0,y_0)}{\partial x} = z_0$ and that you want to solve $\frac{\partial F (x,y)}{\partial x} = z$ for $y$ and $z$ in a neighborhood of $y_0$ and $z_0$, respectively. This you may do if the linearized equation is solvable and the partial derivative is reasonably smooth. In particular, the slope (if in one variable) $$a = \frac{\partial}{\partial x} \frac{\partial F (x,y)}{\partial x} (x_0,y_0) \neq 0$$ (or in higher dimension should be an invertible matrix). In that case you may solve and obtain as solution $x=G(y,z)$ in some (small) neighborhood of $(y_0,z_0)$. For the derivatives you may use that $$ \frac{\partial F }{\partial x}(G(y,z),y) - z = 0$$ and take derivatives with respect to $y$ and $z$. For example taking a $z$-derivative: $$ \frac{\partial^2 F }{\partial x^2}(G(y,z),y)\frac{\partial G(y,z)}{\partial z} - 1 = 0$$