This is the transformation which is needed to convert a reading x' in the K' system of coordinates to an x value in the K system, if K' moves with a speed of v relative to the K system. According to classical Galileian / Newtonian mechanics: x' = x - vt.
We are considering motion only along the x axis. y' and z' remain the same as y and z, respectively.

A light signal along the positive x axis ( to the right ) of frame K is transformed according to the
equation x = ct, or x - ct = 0                                                                                            (1)

Relative to K' the transformation is x' - ct' = 0                                                                     (2)

Space-time points ( called events ) which satisfy (1) must also satisfy (2).

                                    i.e.   where l is a constant                              (3)

In the negative direction where m is a constant                                     (4)

Add (4) to (3)     

                                i.e.                                                                       (5 i)

        where and .

      Equation (3)

      Equation (4)

Subtract (4) from (3)

                      So that

                                 i.e.

                         ..So that                                                                      (5 ii)

The problem now is to find the values of a and b. For the origin of K', x' = 0
x' = ax - bct. ( Equation (5 i) ), becomes ax - bct = 0.

                                                                                                         (6)

                                    and or , and .

v is the relative velocity of the two systems of co-ordinates.

The Principle of Relativity states: Relative to K the length of a unit measuring rod at rest to K', must be exactly the same as the length of the rod relative to K' of a unit measuring rod at rest to K.

Insert a particular value of t ( time of K ), e.g. t = 0,

                                     Then (5 i) becomes x' = ax - bc(0),

                                                        So that x' = ax

If x' increases by 1 unit ( infinitesimally small ) ax = 1 so that                           (7)

Relative to K' where t' = 0 , eliminating t from equations (5 i) and (5 ii)

we get: bct = ax - x' and act = ct' + bx .

            Thus and so that .

                                so that….. .

But and from (6)

                                                     ie

                                             so that .

Thus points on the x-axis separated by distance 1 relative to K will be represented by the distance

x' = a ( 1 - v² / c² )1 and this must be equal to ( from (7) )

. Therefore                                                           (8)

Equations (6) and (8) determine the values of a and b . Insert these values of
a and b in equation (5 i) :    

                                           
                        The Lorentz transformation for events on the x-axis.

The time t' can be derived from:

It equals the classical transformation divided by