Linear active-power-only power flow approximations are pervasive in the planning and control of power systems. However, AC power systems are governed by a system of nonlinear nonconvex power flow equations. Existing linear approximations fail to capture key power flow variables, including reactive power and voltage magnitudes, both of which are necessary in many applications that require voltage management and AC power flow feasibility. This paper proposes novel linear-programming models (the LPAC models) that incorporate reactive power and voltage magnitudes in a linear power flow approximation. The LPAC models are built on a polyhedral relaxation of the cosine terms in the AC equations as well as Taylor approximations of the remaining nonlinear terms. Experimental comparisons with AC solutions on a variety of standard IEEE and Matpower benchmarks show that the LPAC models produce accurate values for active and reactive power, phase angles, and voltage magnitudes. The potential benefits of the LPAC models are illustrated on two “proof-of-concept” studies in power restoration and capacitor placement.