Solutions in 1-D

Generally, solving multi-D equations is much harder, so we'll start with the 1-D case first...
By writing f(x) = 0 we have reduced the problem to solving for the roots of f.
In 1-D it is usually possible to trap or bracket the desired roots and hunt them down.
Typically all root-finding methods proceed by iteration, improving a trial solution until some convergence criterion is satisfied.

Generally, solving multi-D equations is much harder, so we'll start with the 1-D case first...