I'm just learning about relativity, and every equation I see for a galilean transformation of frame $S'$ (moving with uniform velocity in the $x$-direction with respect to frame $S$) is $x'=x-vt$, $y'=y$, $z'=z$, $t'=t$. My question involves the $x'$ term. Don't these equations assume that frames $S$ and $S'$ align together (origins $O$ and $O'$ are the same point) at $t=0$? This assumption is not explicitly stated in my textbooks, and it seems like the general formula should be $x'=x-x_0-vt$, where $x_0$ is the distance (in $x$-direction) the $S'$ frame is away from the $S$ frame (along the $x$-axis) ath $t=0$. This wouldn't change the velocity transformation formula, because $x_0$ is constant. Am I right about this? If so, why doesn't the canonical form for a galilean transformation include this initial displacement term?