LinearSolve(F, R) returns an equivalent linear semi-algebraic system of triangular shape to the input linear semi-algebraic system F. This assume that R is the field of rational numbers.

The output linear triangular semi-algebraic system satisfies the following properties: Firstly, the output constraints are still linear and are sorted in an ascending order according to the largest variable appearing in them. Secondly, the projection of the solutions of input system F onto any lower dimensional space, say the space formed by the smallest  variables, are exactly the solutions of those constraints in the output which only involve the smallest  smallest variables. The algorithm behind this function is a variant of Fourier-Motzkin elimination.

The RegularChains[SemiAlgebraicSetTools][LinearSolve] command was introduced in Maple 16.

For more information on Maple 16 changes, see Updates in Maple 16.