Equation Solving - Overview

Overview

In one general case, we have a situation such as

ƒ (x₁,...,x_n) = c,

where x₁,...,x_n are the unknowns, and c is a constant. Its solutions are the members of the inverse image

ƒ −1 = {(a₁,...,a_n) ∈ T₁×···×T_n | ƒ (a₁,...,a_n) = c},

where T₁×···×T_n is the domain of the function ƒ. Note that the set of solutions can be empty (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

3x + 2y = 21z

with unknowns x, y and z, can be solved by first modifying the equation in some way while keeping it equivalent, such as subtracting 21z from both sides of the equation to obtain

3x + 2y − 21z = 0

In this particular case there is not just one solution to this equation, but an infinite set of solutions, which can be written

{(x, y, z) | 3x + 2y − 21z = 0}.

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. In fact, this particular set of solutions describes a plane in three-dimensional space, which passes through the three points with these coordinates.