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The full derivation is at

http://netbula.com/ydx-SR-lorentz-transformation.pdf

Put it simply, let's assume the general form of the transformation of event (x_A, t_A)

from reference A to B (where B moves at v relative to A) is

x_B = r * (x_A – v * t_A )

switching the references around, A moves at -v relative to B, we must have

x_A = r * (x_B + v * t_B )

From these two equations, we get

t_B = r * t_A + x_A * (1 – r*r) /(r*v)

Now, if we want space and time be treated symmetrically by our nature, we have

- r * v = (1-r*r)/ (r*v)

And we find

r = 1 /sqrt (1- v * v)

For analysis, see http://netbula.com/ydx-SR-lorentz-transformation.pdf