If $$\begin{gathered} {f_{11}}({x_1}) + {f_{12}}({x_2}) = {c_1}, \hfill \\ {f_{21}}({x_1}) + {f_{22}}({x_2}) = {c_2}, \hfill \\ \end{gathered} $$
where $c_1$ and $c_2$ are real constants, the $f_{ij}$'s are real functions of one real variable and $x_1$ and $x_2$ are the unknowns, why does that imply that both $x_1$ and $x_2$ are constant? Is it simply because there are two equations and two unknowns?
EDIT: The functions $f_{ij}$ are not constant functions.