  We consider the K3 surfaces that arise as double covers of the elliptic
modular surface of level 5, $R_{5,5}$. Such surfaces have a natural elliptic
fibration induced by the fibration on $R_{5,5}$. Moreover, they admit several
other elliptic fibrations. We describe such fibrations in terms of linear
systems of curves on $R_{5,5}$. This has a major advantage over other methods
of classification of elliptic fibrations, namely, a simple algorithm that has
as input equations of linear systems of curves in the projective plane yields a
Weierstrass equation for each elliptic fibration. We deal in detail with the
cases for which the double cover is branched over the two reducible fibers of
type $I_5$ and for which it is branched over two smooth fibers, giving a
complete list of elliptic fibrations for these two scenarios.
