[Speedystar777]

I was trying to prove the combined gas law to myself, but ended up with an equation that contradicts Boyle’s Law. What logical fallacies did I make?

[Meter]

Your algebra is obviously correct. Your problem is in your interpretation of the laws. Remember what each law assumes as an initial condition and how that differs between the laws.

[Babcock_Hall]

I think learning by doing one's own derivations is a fine way to proceed, and making mistakes is an important part of the learning process.

[MNIO]

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