Lorentz Transform Derivation questions

[kwestion]

I'm trying to follow along with Simple Derivation of the Lorentz Transformation, but am having some hurdles.

I'll be referring to step (5) which states:

x'=ax-bct
ct'=act-bx​

In paragraph marked 6, I see that the author tries to get eqn (5) to describe motion of the K' frame. This is an important move, but not understood. Up until that point, I believe x' has been a description of the position of light on the frame K' with x' having rules of motion that include x'=ct'. x'=ct' suggests to me that x' is at the K' origin for only a moment when t'=0, but the author states that:

For the origin of k' we have permanently x' = 0[...]​

I don't understand the "permanence" here. Does x' linger at the K' origin? Did x' change meaning? Is it poor notation? Is it that since t'=0 is the only valid moment* for (5) that the state of that moment constitutes a permanent state for (5)? Is there a better description of why (5) begins to be used to track the motion of the frame? I don't see how the position of x' helps understand the movement of K' here. * "The only valid moment" is an unconfirmed assumption on my part. (5) was derived from equations like x-ct=0 and x+ct=0 (inferred) and x'-ct'=0 and x'+ct'=0 (inferred). Upon combining equations in (5), I think all former conditions need to be satisfied by any x, t, x', or t' used with (5). That is, valid x,t,x',t' must not contradict any of: x-ct=0, x+ct=0, x'-ct'=0, or x'+ct'=0, which implies that x=0, t=0, x'=0, and t'=0. What perspective am I missing?

 

[bcrowell]

This is just a matter of definition. The x' coordinate in the frame K' is defined by measuring distances from some inertially moving object that is at rest in the K' frame.