How can we combine all the three gas laws into a single ideal gas equation?
It is indeed the case that the rearrangement of the different gas laws is not very satisfactory as it employs the different variables without any constraints. This issue has been addressed by S. Levine in J. Chem. Educ., 1985, 62 (5), p 399, but in the case you do not have access to this reference, I'll shortly repeat his arguments here. 

As $P$, $V$ and $T$ are thermodynamic sate functions, we may write the total differential of $T$ as

$$\mathrm{d}T=\left (\frac{\partial T}{\partial V} \right )_P \mathrm{d}V + \left (\frac{\partial T}{\partial P} \right )_V\mathrm{d}P \equiv X\mathrm{d}V+ Y\mathrm{d}P$$

since this differential must be exact, the following relation must hold

$$\left (\frac{\partial X}{\partial P} \right )_V=\left (\frac{\partial Y}{\partial V} \right )_P$$

We can find simple functions satisfying this condition
$$X=cP\text{ and }Y=cV$$
where $c$ is a constant that we may choose to be $c=1/R$ for later convenience.  Substitution of these functions in the total differential gives
$$\mathrm{d}T=\frac{1}{R}\left (P\mathrm{d}V +  V\mathrm{d}P\right )$$
and integrating on both sides from 0K to T yields
$$T-T_0=\frac{1}{R}\left [(PV)_T-(PV)_{T_0} \right ]$$
and since $T_0=0$ and $PV=0$ at $\pu{0K}$ we arrive at the ideal gas law.