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Is there general way to do "reverse", is there a subject exist for any function that is possible to "reverse" and for solving these problem for all types of functions or equations?

Example:What is a fast way to write $a,b,c$ in terms $x,y,z$ (I call these kind of problem "reverse" problems) if $$x=\frac{(1-b)(1-c)}{(a-b)(a-c)},\quad y=\frac{(1-c)(1-a)}{(b-c)(b-a)},\quad z=\frac{(1-a)(1-b)}{(c-a)(c-b)}$$

By the way, Is all sort of equation or function have their corresponding "reversal" equation or function, if not, how do we classify all sort of equation or functions into different groups based on their properties?

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Well, for starters $x+y+z=1$ so you won't be able to invert arbitrary triples $(x,y,z)$. –  Ted Apr 16 '12 at 3:39

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