Many answers are possible to show the existence of dark matter
including:

- the rotation curves of spiral galaxies remain flat well outside the
optical extent of the galaxy. Unexpected from Kepler's law.

- Clusters of galaxies: mass determinations from the virial theorem, an
application of the equation of hydrostatic equilibrium and X-ray
observation, and gravitational lensing all imply a high M/L ratio.

- Applications of virial-type analyses from the solar neighborhood, to
the local group, up to scales of superclusters all yield high M/L
rations.

- the observations of massive structure at high-z imply a
(non-baryonic) component of the mass content of the universe (otherwise
the observed fluctuations in dt/T in the microwave background would be
too high).

- something to note is that Big Bang nucleosynthesis implies that the
dark matter must be baryonic, since e.g., M/L ratios on large scales
imples the Omega -> 0.3 but BBN (a well tested theory) implies the
fraction of the critical density in baryons is ~0.05.