<s>[INST] Find all functions $F(x)$ for which $F (x) + F ((x − 1)/x) = 1 + x$ 
Let $F (x)$ be the real-valued function defined for all real $x$ except for $x = 0$ and $x = 1$ and satisfying the functional equation $F (x) + F ((x − 1)/x) = 1 + x$. Find $F (x)$.

This functional equation looks like I could do an inverse substitution. Meaning, let $x = \dfrac{x-1}{x}$ then we have $F \left(\dfrac{x-1}{x} \right)+F\left( \dfrac{1}{x-1}\right) = \dfrac{2x-1}{x}$. Thus, $1+x-F(x) =  \dfrac{2x-1}{x}-F\left( \dfrac{1}{x-1}\right)$. I am not sure how to proc
 [/INST] Solve the system of questions:
$$
\begin{align}
F(x) + F\left(\frac{x-1}{x}\right) &= 1+x \\
F\left(\frac{x-1}{x}\right) + F\left(\frac{1}{1-x}\right) &= \frac{2x-1}{x} \\
F\left(\frac{1}{1-x}\right) + F(x) &= \frac{2-x}{1-x}
\end{align}
$$
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