Is it the case that the equation Var(a + bY) = bVar(Y) is always true? If so, how can you prove that this equation is always true?
It will be $Var(a + bY) = b^2 Var(Y)$ and yes it is always true... You can proceed in a way like $Var(k) = 0$ if $k$ is a constant. Now if you want to work with a constant then it is purely fixed at some point only so location is fixed i.e. the $E(k) = k$. Now as that point is already fixed so it does not have any deviation from its mean or location point i.e. the variance is zero.