Both the CO(2-1) and CO(1-0) lines are used to trace the mass of molecular gas in galaxies. Translating the molecular gas mass estimates between studies using different lines requires a good understanding of the behaviour of the CO(2-1)-to-CO(1-0) ratio, $R_{21}$. We compare new, high quality CO(1-0) data from the IRAM 30-m EMPIRE survey to the latest available CO(2-1) maps from HERACLES, PHANGS-ALMA, and a new IRAM 30-m M51 Large Program. This allows us to measure $R_{21}$ across the full star-forming disc of nine nearby, massive, star-forming spiral galaxies at 27 (${sim} 1{-}2$ kpc) resolution. We find an average $R_{21} = 0.64pm0.09$ when we take the luminosity-weighted mean of all individual galaxies. This result is consistent with the mean ratio for disc galaxies that we derive from single-pointing measurements in the literature, $R_{rm 21, lit}~=~0.59^{+0.18}_{-0.09}$. The ratio shows weak radial variations compared to the point-to-point scatter in the data. In six out of nine targets the central enhancement in $R_{21}$ with respect to the galaxy-wide mean is of order $sim 10{-}20%$. We estimate an azimuthal scatter of $sim$20% in $R_{21}$ at fixed galactocentric radius but this measurement is limited by our comparatively coarse resolution of 1.5 kpc. We find mild correlations between $R_{21}$ and CO brightness temperature, IR intensity, 70-to-160$ mu$m ratio, and IR-to-CO ratio. All correlations indicate that $R_{21}$ increases with gas surface density, star formation rate surface density, and the interstellar radiation field.
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