The Plane command in the Student MultivariateCalculus package will generate the representation of a plane from a list of three points it contains. The return already has had the equivalent of the primparts operator applied. It was in an attempt to understand just what the primparts command did that I spent time looking at these calculations. I ended up with the following code to generate all the planes induced by the given list L.
for k from 1 to nops(L) do
The Plane command creates a "plane object," which is, I believe, a module containing all the information about the plane. To see an equation for the plane, apply the GetRepresentation command. This returns an equation in the form a x+...=d, so some jiggering is needed to move everything to the left. Then, multiplication by the signum of the lead coefficient puts the equation in the desired form (as suggested by vv). Kitonum suggests a different strategy to make the lead coefficient positive.
By printing every equation on a separate line, it was easier to compare equations. I noticed that every equation contained the variable x. I added [[0,0,0],[1,0,0],[0,0,1]] to L and was happy to see that my code returned y=0 for that case.
I find that if I don't explore the various bits of code provided in the responses to the questions on MaplePrimes, I really don't learn much just by reading through these replies.