The algebra/calculus is not too hard for this. Considering only lift-dependent drag and profile drag as a function of speed v and weight W, the drag curve (in arbitrary units) looks like:
D(v, W) = W^2/v^2 + v^2
So e.g. for W = 1, the minimum of 2 is at v = 1.
More generally, the minimum is D_min = 2W at v_min = sqrt(W). So if you increase W, the curve shifts right and up.
Interesting that D_min scales with W. That's consistent with the idea of a "best L/D" AoA, at which D simply scales with L (which is W).
(Introducing drag with other than a quadratic or zero dependence on lift will change the results a bit, but the effect will be similar.)