This paper deals with the incremental detection of implicit equalities using the revised simplex method. This algorithm is more efficient and more suitable to practical problems than the tableau method usually applied in constraint logic programming. We describe and discuss the adaptation to the revised simplex of three approaches: the CLP (R), the Prolog III, and the quasi-dual one. All of these have been integrated into the constraint logic programming language Athena based on a revised simplex method over exact-precision rationals. This system is used to compare these methods on a set of typical CLP problems over linear constraints.