Equation 5 Equation 8 Equation 7 Lorentz transformation
Substituting expressions for 



d

x
′



{\displaystyle dx'}

 and 



d

t
′



{\displaystyle dt'}

 from Equation 5 into Equation 8, followed by straightforward mathematical manipulations and back-substitution from Equation 7 yields the Lorentz transformation of the speed 



u


{\displaystyle u}

 to 




u
′



{\displaystyle u'}

:
