  We construct a cosmological model from the inception of the
Friedmann-Lem\^aitre-Robertson-Walker metric into the field equations of the
f(R,L_m) gravity theory, with R being the Ricci scalar and L_m being the matter
lagrangian density. The formalism is developed for a particular f(R,L_m)
function, namely R/16\pi +(1+\sigma R)L_{m}, with \sigma being a constant that
carries the geometry-matter coupling. Our solutions are remarkably capable of
evading the Big-Bang singularity as well as predict the cosmic acceleration
with no need for the cosmological constant, but simply as a consequence of the
geometry-matter coupling terms in the Friedmann-like equations
