In this paper, we consider the manifold that describes all feasible power flows in a power system as an implicit algebraic relation between nodal voltages (in polar coordinates) and nodal power injections (in rectangular coordinates). We derive the best linear approximant of such a relation around a generic solution of the power flow equations. Our linear approximant is sparse, computationally attractive, and preserves the structure of the power flow. Thanks to the full generality of this approach, the proposed linear implicit model can be used to obtain a fast approximate solution of a possibly unbalanced three phase power system, with either radial or meshed topology, and with general bus models. We demonstrate how our approximant includes standard existing linearizations, we validate the quality of the approximation via simulations on a standard testbed, and we illustrate its applicability with case studies in scenario-based optimization and cascading failures.