SolveAlways[eqns, vars] gives the values of parameters that make the equations eqns valid for all values of the variables vars.
Equations are given in the form lhs == rhs.
Simultaneous equations can be combined either in a list or with &&.
A single variable or a list of variables can be specified.
Example: SolveAlways[a x + b == 0, x] .
SolveAlways works primarily with linear and polynomial equations.
SolveAlways produces relations between parameters that appear in eqns, but are not in the list of variables vars.
SolveAlways[eqns, vars] is equivalent to Solve[!Eliminate[!eqns, vars]].
See The Mathematica Book: Section 3.4.10.
See also: Eliminate, Solve, Reduce, PolynomialReduce.