\(\frac{(P1)(V1)}{T1}=\frac{(P2)(V2)}{T2}\) is directly related to the ideal gas law.
The ideal gas law states that:\(PV=nRT\). Rewriting this gives:\(\frac{PV}{T}=nR\). R is the universal gas constant, so its value is fixed. For a given sample, n is the number of moles of the gas, so its value would be fixed too. This means that the value of \(\frac{PV}{T}\) would be a fixed constant as well. Therefore, whatever the initial value \(\frac{(P1)(V1)}{T1}\) is, it should be equal to the final value \(\frac{(P2)(V2)}{T2}\).
