Equation solving
Overview

One general form of an equation is

where f is a function, x,..., x are the unknowns, and c is a constant. Its solutions are the elements of the inverse image (fiber)

where D is the domain of the function f. The set of solutions can be the empty set (there are no solutions), a singleton (there is exactly one solution), finite, or infinite (there are infinitely many solutions).

For example, an equation such as

with unknowns x, y and z, can be put in the above form by subtracting 21 z from both sides of the equation, to obtain

In this particular case there is not just one solution, but an infinite set of solutions, which can be written using set builder notation as

One particular solution is x = 0, y = 0, z = 0. Two other solutions are x = 3, y = 6, z = 1, and x = 8, y = 9, z = 2. There is a unique plane in three-dimensional space which passes through the three points with these coordinates, and this plane is the set of all points whose coordinates are solutions of the equation.
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