Scaling analysis is used to derive approximations of magnetohydrodynamics with self-consistent leading-order dynamics under general conditions of anisotropy. Both incompressible and weakly compressible limits are considered. The horizontal magnetic and velocity fields obey dynamics given by a reduced closed set of equations, but the vertical components have different decoupled dynamics in the different regimes. Conservation laws are also discussed. It is shown that the reduced equations are not self-consistent unless either the ratio of vertical to horizontal length scales is large or the fluid is gravitationally stratified and the ratio of length scales is small. New equations are derived for a rotating stratified plasma, which are an extension of the quasigeostrophic equations of neutral fluids.