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We present the first results from “Surveying the Whirlpool at Arcseconds with NOEMA” (SWAN), an IRAM Northern Extended Millimetre Array (NOEMA)+30 m large program that maps emission from several molecular lines at 90 and 110 GHz in the iconic nearby grand-design spiral galaxy M 51 at a cloud-scale resolution (∼3″ = 125 pc). As part of this work, we have obtained the first sensitive cloud-scale map of N₂H⁺(1–0) of the inner ∼5  × 7 kpc of a normal star-forming galaxy, which we compared to HCN(1–0) and¹²CO(1–0) emission to test their ability in tracing dense, star-forming gas. The average N₂H⁺-to-HCN line ratio of our total FoV is 0.20 ± 0.09, with strong regional variations of a factor of ≳2 throughout the disk, including the south-western spiral arm and the center. The central ∼1 kpc exhibits elevated HCN emission compared to N₂H⁺, probably caused by AGN-driven excitation effects. We find that HCN and N₂H⁺are strongly super-linearily correlated in intensity (ρ_Sp ∼ 0.8), with an average scatter of ∼0.14 dex over a span of ≳1.5 dex in intensity. When excluding the central region, the data are best described by a power law of an exponent of 1.2, indicating that there is more N₂H⁺per unit HCN in brighter regions. Our observations demonstrate that the HCN-to-CO line ratio is a sensitive tracer of gas density in agreement with findings of recent galactic studies utilising N₂H⁺. The peculiar line ratios present near the AGN and the scatter of the power-law fit in the disk suggest that in addition to a first-order correlation with gas density, second-order physics (such as optical depth, gas temperature) or chemistry (abundance variations) are encoded in the N₂H⁺/¹²CO, HCN/¹²CO, and N₂H⁺/HCN ratios.