  We present new analytic solutions for one-zone (fully mixed) chemical
evolution models and explore their implications. In contrast to existing
analytic models, we incorporate a realistic delay time distribution for Type Ia
supernovae (SNIa) and can therefore track the separate evolution of
$\alpha$-elements produced by core collapse supernovae (CCSNe) and iron peak
elements synthesized in both CCSNe and SNIa. In generic cases, $\alpha$ and
iron abundances evolve to an equilibrium at which element production is
balanced by metal consumption and gas dilution, instead of continuing to
increase over time. The equilibrium absolute abundances depend principally on
supernova yields and the outflow mass loading parameter $\eta$, while the
equilibrium abundance ratio [$\alpha$/Fe] depends mainly on yields and
secondarily on star formation history. A stellar population can be metal-poor
either because it has not yet evolved to equilibrium or because high outflow
efficiency makes the equilibrium abundance itself low. Systems with ongoing gas
accretion develop metallicity distribution functions (MDFs) that are sharply
peaked, while "gas starved" systems with rapidly declining star formation have
broadly peaked MDFs. A burst of star formation that consumes a significant
fraction of a system's available gas can temporarily boost [$\alpha$/Fe] by
0.1-0.3 dex, a possible origin for rare, $\alpha$-enhanced stars with
intermediate age or high metallicity. Other sudden transitions in system
properties can produce surprising behavior, including backward evolution of a
stellar population from high metallicity to low metallicity. An Appendix
provides a user's guide for calculating enrichment histories, [$\alpha$/Fe]
tracks, and MDFs for a wide variety of scenarios, including flexible forms of
star formation history.
