Deriving combined gas law from Boyle's and Charles' laws

You have to realize first that Charles' law is the change in volume with respect to temperature for constant pressure while Boyle's law is the change in volume with respect to pressure for constant temperature. So when you combine them, you need to account for these

If I take a gas of volume $V_1$, pressure $P_1$ and temperature $T_1$ and let it change have a state $(V_*,\,P_2,\,T_1)$, then via Boyle's law, $$ P_1V_1=P_2V_*\tag{1} $$ Then keeping this constant pressure we move to state $(V_2,\,P_2,\,T_2)$ via Charles' law, $$ \frac{V_*}{T_1}=\frac{V_2}{T_2}\tag{2} $$ Solving for $V_*$ in both (1) and (2) then equating them, we get $$ \frac{P_1V_1}{T_1}=\frac{P_2V_2}{T_2} $$ We've changed the pressure, volume and temperature of the gas but still find their product equal, suggesting that the relation is constant: $$ \frac{PV}{T}=k $$