Compilation to Boolean satisfiability has become a powerful paradigm for solving artificial intelligence problems. However, domains that require metric reasoning cannot be compiled efficiently to satisfiability even if they would otherwise benefit from compilation. We address this problem by combining techniques from the artificial intelligence and operations research communities. In particular, we introduce the LCNF (Linear Conjunctive Normal Form) representation that combines propositional logic with metric constraints. We present LPSAT (Linear Programming plus SATisfiability), an engine that solves LCNF problems by interleaving calls to an incremental Simplex algorithm with systematic satisfaction methods. We explore several techniques for enhancing LPSAT's efficiency and expressive power by adjusting the interaction between the satisfiability and linear programming components of LPSAT. Next, we describe a compiler that converts metric resource planning problems into LCNF for processing by LPSAT. Finally, the experimental section of the paper explores several optimisations to LPSAT, including learning from constraint failure and randomised cutoffs.