TABLE OF CONTENTS In this chapter we discuss existence and uniqueness of solutions to various equations and show how to use Maple to find solutions. We deal mainly with polynomial equations in one unknown and add only some basic facts about systems of linear equations.
Given a function f, solving the equation [1]
| (8.1) |  |
means finding all x in dom f which satisfy Equation (8.1). Any x which satisfies Equation (8.1) is called a solution of Equation (8.1). In this chapter we assume that dom f is always part of 
or 
and it will be clear from the context which set is meant. Solving an equation like (8.1) usually consists of a chain of implications, starting with the equation itself and ending with an equation (or equations) of the form x=a. For instance:
| (8.2) |  |
For greater clarity we printed the implication signs, but implications are always understood automatically. It is a good habit always to check the solution by substituting the found value back into the original equation. This is not a mere verification of the calculations, it has a far more fundamental reason. Solving an equation starts with an assumption that the equation is satisfied for some x, in other words, it is assumed that a solution exists. If it does not then the chain of implication can lead to a wrong result. [2] Consider the following example [3]
| (8.3) |  |
Substituting 0 in the original equation...
