When you write down a set of simultaneous equations in the Wolfram Language, you are specifying a collection of constraints between variables. When you use Solve, you are finding values for some of the variables in terms of others, subject to the constraints represented by the equations.
|Solve[eqns,vars,elims]||find solutions for vars, eliminating the variables elims|
|Eliminate[eqns,elims]||rearrange equations to eliminate the variables elims|
In some cases, you may want to construct explicitly equations in which variables have been eliminated. You can do this using Eliminate.
As a more sophisticated example of Eliminate, consider the problem of writing in terms of the "symmetric polynomials" and .
In dealing with sets of equations, it is common to consider some of the objects that appear as true "variables", and others as "parameters". In some cases, you may need to know for what values of parameters a particular relation between the variables is always satisfied.
|SolveAlways[eqns,vars]||solve for the values of parameters for which the eqns are satisfied for all values of the vars|