Eliminate[eqns, vars] eliminates variables between a set of simultaneous equations.
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: Eliminate[x == 2 + y, y == z, y] .
Variables can be any expressions.
Eliminate works primarily with linear and polynomial equations.
See The Mathematica Book: Section 1.5.7 and Section 3.4.8.
See also: Reduce, SolveAlways, Solve, GroebnerBasis.