That's true, but I'm not sure it's helpful as it leaves you with 1 equation and 2 unknowns.

I am presuming that Vin is known and Vout is not. Even if Vin is not known, you can get the answer as a function of Vin.

What I suggested is

Id = (k/2)(Vgs-Vt)^2 [1] and Vin=Vgs+Id*Rs [2]

Therefore (by [2]) Id = (Vin-Vgs)/Rs

So now with 2 equations for Id, knowing that they must be equal we can solve for Vgs. (which turns out to be done by solving a quadratic equation).

Having established Vgs, everything else falls out using (1) and (2).

I would be curious how you would solve it using your approach. This isn't my area of expertise (I only learned it yesterday

) so maybe there's something very obvious I'm missing.