Understanding Lorentz transformation

Note that the transformations:

begin{aligned}x' &= gamma left(x- v tright) t' &= gammaleft(t - frac{vx}{c^2}right)end{aligned}

have been defined so that when $t=t'=0$, $x=x'=0$, meaning that the clocks are already synchronised. This is the usual convention: the two observers in $S$ and $S'$ synchronise their clocks when their "origins" coincide. This ist, of course, not essential, but it certainly makes the equations a little more compact. I believe it's usually discussed in most introductory derivations of the Lorentz Transformations.

Here is an image of the $(t,x)$ and $(t',x')$ axes.

Note that $t = 0$ is the $x$-axis, and $t' = 0$ is the $x'$-axis. Based on the picture, you can see that the $x$-axis and $x'$-axis only intersect at $x = x' = 0$.

Lorentz transformation are contained in the Poincare transformation where the time coordinate transform as $$t'=gammaleft(t-frac{vx}{c^2}right)+a$$ where $a$ is some constant. So for example if they are separated by a distance $d$ if we set $a=gammafrac{vd}{c^2}$ we see that both of their clock are zero.