Why do scientists use a different method from redshift to find the distance of distant galaxies? The method I am referring to is measuring the brightness of a Type Ia supernova. Does the redshift method even work at these distances and if it doesn't why not? If it does work, why do they prefer to use the supernova method?
 A: The redshift-distance formula $v = H_0 d$, where $v$ is the recession velocity and $d$ is the distance assumes that $H_0$, "the Hubble constant" is constant in time ($H_0$ indicates that this is the value now.). In fact for a decelerating or accelerating universe it is not constant and so a (much) more complicated formula has to be invoked involving the values of the matter density and dark energy density of the universe (which control the dynamics, see for example equation 2 in Riess et al. 1998).
However, the goal is often to actually find the values of these densities, in which case you need to come up with another way of estimating the distance; then measure the recession velocities and fit the complicated formulae to determine the values of the mass and dark energy densities. This is essentially the "supernova cosmology experiment" for which three astronomers were awarded a Nobel prize.
So yes, if you know what the mass and dark energy densities of the universe are then redshift can be used to estimate distance. But if you want to find the mass and energy densities then you use standard candles like type Ia supernovae to give you the distance.
