Suppose that I have $M$ elementary functions $F = \{f_m: \mathbb{R}^N \rightarrow \mathbb{R}\}_{m=1}^M$ and $N$ independent variables $x_1, x_2, \ldots, x_N$. Under what general circumstances can one combine the functions in $F$ with each other and with constants to isolate individual variables, or generate a constant non-trivially?
Some simple examples: if $f_1(x, y) = x + y, f_2(x, y) = 2x$, we can isolate $y$ by $f_1 - \frac{f_2}{2}$. I don't think we can isolate either x nor y in $f_1(x, y) = xy, f_2(x, y) = x + y$.