A long-standing question is whether active galactic nuclei (AGN) vary like Galactic black hole systems when appropriately scaled up by mass. If so, we can then determine how AGN should behave on cosmological timescales by studying the brighter and much faster varying Galactic systems. As X-ray emission is produced very close to the black holes, it provides one of the best diagnostics of their behaviour. A characteristic timescale--which potentially could tell us about the mass of the black hole--is found in the X-ray variations from both AGN and Galactic black holes, but whether it is physically meaningful to compare the two has been questioned. Here we report that, after correcting for variations in the accretion rate, the timescales can be physically linked, revealing that the accretion process is exactly the same for small and large black holes. Strong support for this linkage comes, perhaps surprisingly, from the permitted optical emission lines in AGN whose widths (in both broad-line AGN and narrow-emission-line Seyfert 1 galaxies) correlate strongly with the characteristic X-ray timescale, exactly as expected from the AGN black hole masses and accretion rates. So AGN really are just scaled-up Galactic black holes.