Special relativity - time dilation

[Achmed]

Consider two events that take place at the origin of the frame of an inertial observer $O'$. At times $t_1 ' = 0$ and $t_2 ' = T$. $O'$ moves with a constant speed $v$ w.r.t. another inertial observer $O$.

1. Use the Lorentz-transformations to show that these events occur at $x=vt$ in the frame of $O$, at the times $t_1 = 0$ and $t_2 = \gamma T$. Show furthermore that the events do not occur at the same place in $O$, but at $x_1=0$ and $x_2 = \gamma vT$.

My attempt below. I am very confused, so I have no idea if what I'm doing is even remotely correct:

Since both events take place at the origin of $S'$, we get that $x' = 0$. From here it follows using the Lorentz transformation that $x=vt$.

After this the confusion starts. If we use the Lorentz transformation for time, and we use $t'_1 = 0$, we actually get $t_1 = \dfrac{vx}{c^2}$. But this doesn't correspond with what they ask, right? If we do the same for $t_2' = T$, we get the very same problem. What am I doing incorrectly here?

Using the same tactic above, if I fill in $x' = 0$ and $t_1 = 0$ and $t_2 = \gamma T$, I do get the correct answers for $x_1$ and $x_2$! So, that confuses me..

 

[Orodruin]

You did not do something wrong. It is just that you got a relation between two unknowns, $x_1$ and $t_1$ (the x coordinate corresponding to $t_1$ should be $x_1$). In order to find another equation to help you, use what you just derived from the Lorentz transformation of x, it must be true in particular for $t_1$ and $x_1$.

The alternative is to use the inverse Lorentz transformations from the start, what you are doing essentially corresponds to inverting the transform anyway.

For future reference, you should also leave the homework template when posting your question. It helps ordering your thoughts and structure the post so that we can grasp your problem quicker.