  We present new HCN and HCO$^+$ ($J$=3-2) images of the nearby star-forming
galaxies (SFGs) NGC 3351, NGC 3627, and NGC 4321. The observations, obtained
with the Morita ALMA Compact Array, have a spatial resolution of $\sim$290-440
pc and resolve the inner $R_\textrm{gal} \lesssim$ 0.6-1 kpc of the targets, as
well as the southern bar end of NGC 3627. We complement this data set with
publicly available images of lower excitation lines of HCN, HCO$^+$, and CO and
analyse the behaviour of a representative set of line ratios:
HCN(3-2)/HCN(1-0), HCN(3-2)/HCO$^+$(3-2), HCN(1-0)/CO(2-1), and
HCN(3-2)/CO(2-1). Most of these ratios peak at the galaxy centres and decrease
outwards. We compare the HCN and HCO$^+$ observations with a grid of one-phase,
non-local thermodynamic equilibrium (non-LTE) radiative transfer models and
find them compatible with models that predict subthermally excited and
optically thick lines. We study the systematic variations of the line ratios
across the targets as a function of the stellar surface density
($\Sigma_\textrm{star}$), the intensity-weighted CO(2-1) ($\langle
I_\text{CO}\rangle$), and the star formation rate surface density
($\Sigma_\text{SFR}$). We find no apparent correlation with
$\Sigma_\text{SFR}$, but positive correlations with the other two parameters,
which are stronger in the case of $\langle I_\text{CO}\rangle$. The
HCN/CO-$\langle I_\text{CO}\rangle$ relations show $\lesssim$0.3 dex
galaxy-to-galaxy offsets, with HCN(3-2)/CO(2-1)-$\langle I_\text{CO}\rangle$
being $\sim$2 times steeper than HCN(1-0)/CO(2-1). In contrast, the
HCN(3-2)/HCN(1-0)-$\langle I_\text{CO}\rangle$ relation exhibits a tighter
alignment between galaxies. We conclude that the overall behaviour of the line
ratios cannot be ascribed to variations in a single excitation parameter (e.g.
density or temperature).
