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The Local Group (LG) is an environment accessible to detailed studies of galaxy formation, providing a complement to the early universe. In particular, spectroscopy of resolved stellar populations in the LG provides kinematical and chemical information for individual stars that can be used to infer the history of $L_\star$ galaxies like the Milky Way (MW) and Andromeda (M31). The Gaia revolution in the MW has enabled comprehensive observational studies of the MW's formation history. The exquisite detail in which the MW has been studied is currently not achievable in any other $L_\star$ galaxy. For this reason, the MW is a template for our understanding of galaxy formation. M31 is the only external galaxy that we can currently hope study in a level of detail approaching the MW. Studies of M31 have recently taken on greater significance, given the growing body of evidence that its formation history differs substantially from that of the MW. In an era of limited information about elemental abundances in M31, I have developed a technique to apply spectral synthesis to low-resolution stellar spectroscopy in order to measure abundances for individual giant stars in distant LG galaxies. Through undertaking the largest deep, spectroscopic survey of M31 to date with my collaborators, this has resulted in the first measurements of the elemental abundances in the inner stellar halo and stellar disk of M31, and the largest homogeneous catalog of elemental abundances in M31. With this foundational work, we have opened the doors to detailed studies of the chemical composition of M31. Now, we can begin to ask--and answer--what differences in the elemental abundances of the M31 and the MW imply for our knowledge of galaxy formation in the broader universe.