try it like this... we're starting with V1 at constant T and n... and converting it to V'

.. P1V1 = P2V'

then if you start with V' at T1 and P1 and convert it to V2 at constant P and n

.. V' / T1 = V2 / T2

subbing

.. P1V1 = P2 * (T1 * V2 / T2)

you get to rearrange