  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.
