I am reading about how to generalize the Galilean law of addition of velocities using the Lorentz transformation, but I am confused about one step.
Here, I have the following equations for Lorentz transformation: $$ \overline{t} = \frac{t}{\sqrt{1-v^2}} - \frac{vx}{\sqrt{1-v^2}} $$ $$ \overline{x} = \frac{-vt}{\sqrt{1-v^2}} + \frac{x}{\sqrt{1-v^2}} $$ $$ \overline{y} = y $$ $$ \overline{z} = z $$
Suppose a particle has speed W in the $\overline{x}$ direction of $\overline{O}$, so $\frac{\Delta\overline{x}}{\Delta\overline{t}} = W $. In the other frame O, its velocity is $\frac{\Delta\overline{x}}{\Delta\overline{t}} = W' $. Then, we can deduce $\Delta x$ and $\Delta t$ from the Lorentz transformation. Suppose $\overline{O}$ moves with velocity v with respect to O, then the above equations imply that $$ \Delta x = \frac{\Delta\overline{x} + v\Delta\overline{t}}{\sqrt{1-v^2}} $$ $$ \Delta t = \frac{\Delta\overline{t} + v\Delta\overline{x}}{\sqrt{1-v^2}} $$ I don't see how these two equations are derived. From the equations, I can only derive $$ \Delta \overline x = \frac {\Delta x - v\Delta t}{\sqrt{1-v^2}} $$ $$ \Delta \overline t = \frac {\Delta t - v\Delta x}{\sqrt{1-v^2}} $$
