  Future electromagnetic observations of black holes may allow us to test
General Relativity in the strong-field regime. Such tests, however, require
knowledge of rotating black hole solutions in modified gravity theories, a
class of which does not admit the Kerr metric as a solution. Several rotating
black hole solutions in modified theories have only been found in the
slow-rotation approximation (i.e. assuming the spin angular momentum is much
smaller than the mass squared). We here investigate whether the systematic
error due to the approximate nature of these black hole metrics is small enough
relative to the observational error to allow their use in electromagnetic
observations to constrain deviations from General Relativity. We address this
by considering whether electromagnetic observables constructed from a
slow-rotation approximation to the Kerr metric can fit observables constructed
from the full Kerr metric with systematic errors smaller than current
observational errors. We focus on black hole shadow and continuum spectrum
observations, as these are the least influenced by accretion disk physics, with
current observational errors of about 10%. We find that the fractional
systematic error introduced by using a second-order, slowly-rotating Kerr
metric is at most 2% for shadows created by black holes with dimensionless
spins $\chi\leq0.6$. We also find that the systematic error introduced by using
the slowly-rotating Kerr metric as an exact metric when constructing continuum
spectrum observables is negligible for black holes with dimensionless spins of
$\chi \lesssim 0.9$. Our results suggest that the modified gravity solutions
found in the slow-rotation approximation may be used to constrain realistic
deviations from General Relativity with continuum spectrum and black hole
shadow observations.
