We report measurements of the photon beam asymmetry Sigma for the reactions (gamma) over right arrowp -> p pi(0) and gamma p -> p eta from the GLUEX experiment using a 9 GeV linearly polarized, tagged photon beam incident on a liquid hydrogen target in Jefferson Lab's Hall D. The asymmetries, measured as a function of the proton momentum transfer, possess greater precision than previous pi(0) measurements and are the first. measurements in this energy regime. The results are compared with theoretical predictions based on t-channel, quasiparticle exchange and constrain the axial-vector component of the neutral meson production mechanism in these models.