  Carbon monoxide (CO) emission is the most widely used tracer of the bulk
molecular gas in the interstellar medium (ISM) in extragalactic studies. The
CO-to-H$_2$ conversion factor, $\alpha_{\rm CO}$, links the observed CO
emission to the total molecular gas mass. However, no single prescription
perfectly describes the variation of $\alpha_{\rm CO}$ across all environments
across galaxies as a function of metallicity, molecular gas opacity, line
excitation, and other factors. Using resolved spectral line observations of CO
and its isotopologues, we can constrain the molecular gas conditions and link
them to a variation in the conversion factor. We present new IRAM 30-m 1mm and
3mm line observations of $^{12}$CO, $^{13}$CO, and C$^{18}$O} across the nearby
galaxy M101. Based on the CO isotopologue line ratios, we find that selective
nucleosynthesis and opacity changes are the main drivers of the variation in
the line emission across the galaxy. Furthermore, we estimated $\alpha_{\rm
CO(1-0)}$ using different approaches, including (i) the dust mass surface
density derived from far-IR emission as an independent tracer of the total gas
surface density and (ii) LTE-based measurements using the optically thin
$^{13}$CO(1-0) intensity. We find an average value of $\alpha_{\rm
CO}=4.4{\pm}0.9\rm\,M_\odot\,pc^{-2}(K\,km\,s^{-1})^{-1}$ across the galaxy,
with a decrease by a factor of 10 toward the 2 kpc central region. In contrast,
we find LTE-based values are lower by a factor of 2-3 across the disk relative
to the dust-based result. Accounting for $\alpha_{\rm CO}$ variations, we found
significantly reduced molecular gas depletion time by a factor 10 in the
galaxy's center. In conclusion, our result suggests implications for commonly
derived scaling relations, such as an underestimation of the slope of the
Kennicutt Schmidt law, if $\alpha_{\rm CO}$ variations are not accounted for.
