The relation between X-ray luminosity (L_X) and ambient gas temperature (T) among massive galactic systems is an important cornerstone of both observational cosmology and galaxy-evolution modeling. In the most massive galaxy clusters, the relation is determined primarily by cosmological structure formation. In less massive systems, it primarily reflects the feedback response to radiative cooling of circumgalactic gas. Here we present a simple but powerful model for the L_X-T relation as a function of physical aperture R within which those measurements are made. The model is based on the precipitation framework for AGN feedback and assumes that the circumgalactic medium is precipitation-regulated at small radii and limited by cosmological structure formation at large radii. We compare this model with many different data sets and show that it successfully reproduces the slope and upper envelope of the L_X-T-R relation over the temperature range from ~0.2 keV through >10 keV. Our findings strongly suggest that the feedback mechanisms responsible for regulating star formation in individual massive galaxies have much in common with the precipitation-triggered feedback that appears to regulate galaxy-cluster cores.
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