  Carbon monoxide (CO) provides crucial information about the molecular gas
properties of galaxies. While $^{12}$CO has been targeted extensively,
isotopologues such as $^{13}$CO have the advantage of being less optically
thick and observations have recently become accessible across full galaxy
discs. We present a comprehensive new dataset of $^{13}$CO(1-0) observations
with the IRAM 30-m telescope of the full discs of 9 nearby spiral galaxies from
the EMPIRE survey at a spatial resolution of $\sim$1.5kpc. $^{13}$CO(1-0) is
mapped out to $0.7-1r_{25}$ and detected at high signal-to-noise throughout our
maps. We analyse the $^{12}$CO(1-0)-to-$^{13}$CO(1-0) ratio ($\Re$) as a
function of galactocentric radius and other parameters such as the
$^{12}$CO(2-1)-to-$^{12}$CO(1-0) intensity ratio, the 70-to-160$\mu$m flux
density ratio, the star-formation rate surface density, the star-formation
efficiency, and the CO-to-H$_2$ conversion factor. We find that $\Re$ varies by
a factor of 2 at most within and amongst galaxies, with a median value of 11
and larger variations in the galaxy centres than in the discs. We argue that
optical depth effects, most likely due to changes in the mixture of
diffuse/dense gas, are favored explanations for the observed $\Re$ variations,
while abundance changes may also be at play. We calculate a spatially-resolved
$^{13}$CO(1-0)-to-H$_2$ conversion factor and find an average value of
$1.0\times10^{21}$ cm$^{-2}$ (K.km/s)$^{-1}$ over our sample with a standard
deviation of a factor of 2. We find that $^{13}$CO(1-0) does not appear to be a
good predictor of the bulk molecular gas mass in normal galaxy discs due to the
presence of a large diffuse phase, but it may be a better tracer of the mass
than $^{12}$CO(1-0) in the galaxy centres where the fraction of dense gas is
larger.
