special relativity - A simple coordinate transformation

I'm currently taking my first shot at reading Einstein's 'On The Electrodynamics' (with plenty of mathematical background). With a few pictures, everything has been crystal clear to my intuition, up till:

If we place $x'=x-vt$, it is clear that a point at rest in the system $k$ must have a system of values $x',y,z$, independent of time.

This is, of course, talking about the scenario in which a coordinate frame $k$ is moving at constant velocity along the positive $x$-axis of a stationary frame $K$. I would imagine that a point at rest in $k$ would have coordinates $(x+vt,y,z)$, and all my intuition says that we should set $x'=x+vt$ in order to have the stationary point in $k$ become $(x',y,z)$. What exactly is wrong with this logic?

Answer

The point at rest in $k$ moves as $(x(t),y(t),z(t)) = (x_0+vt,y,z)$ in the frame in which $k$ itself moves with $v$ in positive $x$-direction. Thus, we have to set $x'(t)=x(t)-vt = x_0 +vt-vt$ to obtain a stationary $x'(t) = x_0$ coordinate for a tupel $(x',y,z)$.