Fixed points and system of equations

Question

Suppose that $F : \mathbb{R}^n \rightarrow \mathbb{R}^n$. A fixed-point of function $F$ is such that $$ F(x) = x. $$ We can represent that fixed point as the solution to the system $$ F(x) - x = 0. $$ My question is whether all systems of equations have a fixed-point representation. My intuition is that it depends on whether the system is "separable" in the sense that given a system $$ G(x) = 0, $$ we can find a function $F$ such that $G(x) = F(x) - x$. Is that so? Are there any conditions on $G$ that guarantee that?