Avogadro's hypothesis states that equal volumes of all gases under the same conditions of temperature and pressure contain equal number of molecules.
\(V\propto n\)
\(\frac{V_1}{n_1}=\frac{V_2}{n_2}\) = Constant
Where V₁ and n₁ are the volume and number of moles of a gas and V₂ and n₂ are the different set of values of volume and number of moles of the same gas.
A better example to illustrate Avogadro's hypothesis is to observe the effect of pumping more gas into a balloon. When more gas molecules (particularly CO₂) are passed, the volume of the balloon increases. The pressure and temperature stay constant as the balloon inflates, so the increase in volume is due to the increase in the quantity of gas inside the balloon.

Gay Lussac's law:
At constant volume, the pressure of a fixed mass of a gas is directly proportional .to temperature.
\(P\propto T\) (or) \(P\over T\) = Constant
It can be graphically represented as shown here:

Lines in the pressure vs temperature graph are known as isochores ( constant volume) of a gas.

(i) Variation of volume of the gas sample with temperature at constant
(ii) Each line (iso bar) represents the variation of volume with temperature at certain pressure. The pressure increases from P₁ to P₅.
(iii) i.e. P₁ < P₂ < P₃ < P₄ < P₅ When these lines are extrapolated to zero volume, they intersect at a temperature of -273.15°C.
(iv) All gases are becoming liquids if they are cooled to sufficiently low temperatures.
(v) In other words, all gases occupy zero volume at absolute zero. So the volume of a gas can be measured over only a limited temperature range.

Charles' law states that for a fixed mass of a gas at constant pressure, the volume is directly proportional to temperature (K).
\(V\propto T\) (or)  = \(\frac{V}{T}\) Constant

Volume vs Temperature:
If a balloon is moved from an ice water bath to a boiling water bath, the gas molecules inside move faster due to increased temperature and hence the volume increases.

The pressure-density relationship can be derived from the Boyle's law as shown below.
P₁V₁ = P₂V₂ (Boyle's law)
\(P_1\frac{m}{d_1}-P_2\frac{m}{d_2}\)
Where 'm' is the mass, d₁ and d₂ are the densities of gases at pressure P₁ and P₂ 
\(\frac{P_{1}}{d_{1}}=\frac{P_{2}}{d_{2}}=\)  .....(1)
ln other words,the density of a gas is directly proportional to pressure \((P \propto d)\)