The explanations are written in the attached pdf file.

The two web pages I use as references for formula are
http://en.wikipedia.org/wiki/Lorentz_transformation
and
http://www.wbabin.net/hamdan/hamdan4.htm
However you may not necessarily need them as you likely have all the formula in your textbook(s) or lecture notes.

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Here is the plain TEX source

We shall use the notations
$$
beta = {vover c}
eqno(0.1)
$$
for the dimensionless speed, and
$$
gamma = {1oversqrt{1-beta^2}}
eqno(0.1)
$$
for the Lorentz factor.

bf Q.1rm

Lorentz transformation for the time is
$$
t' = gammal( t - {vxover c^2} r).
eqno(1.1)
$$
As we are requested to have $t'_A = t'_B$ for the two events, from equation (1.1) this requirement

means
$$
gammal( t_A - {vx_Aover c^2} r) = gammal( t_B - {vx_Bover c^2} r).
eqno(1.2)
$$
From equation (1.2) we find the requested difference in times of the events in frame S:
$$
t_A-t_B = {vover c^2} l(x_A-x_Br).
eqno(1.3)
$$

bf Q.2rm

Suppose a particle at $x_1'=0$ is created at $t_1'=0$,
stays at $x_2'=0$ and decays at $t_2' = tau = 1~mu s$ (microsecond).
From the Lorentz transformation back from frame S' to frame S (the same formula as usually ...

Solution Summary

The following posting answers questions about inertial frames.

An event occurs in inertial frame S with coordinates: x=75m, y=18m, z=4m, t=2.0xE-5sec. The inertial frame S' moves in the +x direction at v= .85c. ...

... side is unity from the third postulate, and hence s(L)2 = 1 ... It follows that the set of coordinate transformations between inertial frames (using agreed ...

... inital velocities Vo (m/s): ... To find the path is the inertial frame, we recall that to convert from the rotational frame to the inertial frame we have the ...

... Answer: Data: m = 980 kg v = 20 m/s radius r ... his own is said to be in non-inertial frame of reference. ... Forces which are observed in both the frames of reference ...

...frame of reference moves at a constant speed, relative to other inertial frames of reference ... between the two frames. ... v = 0.9c = 0.9 x 3 x 108 = 2.700 x 108 m/s. ...

... E and the magnetic field B in two different inertial frames. ... E*B is the same for all inertial observers. ... 2: Let us assume frame S' moves along xx' (common to S...

... In the non-rotating frame, the Earth rotates below a ... K/P = 4.9*107/ 8*103 = 0.6125 * 104 s = 6125 Seconds ... has a mass of 44,000 kg, a rotational inertia about a ...

... of her arms, the skater decreases her moment of inertia by 50 ...s c. 9.0 rad/s d. 12 rad/s e. 18 rad/s. ... constant = 53 N/m) is held in place by the equipment frame. ...

... Ã = hc/ ¦Ë = 90,000 x 1.6 * 10^-19 h = 6.6 x 10^-34 Js and c = 3 x 10^8 m/s Substituting these ... are simultaneous in all inertial reference frames. ...

... is to be considered in addition to any other actual force(s) acting on ... lower block as seen by an observer from the ground (ie inertial frame of reference ...