I am confused between Raoult's law and Dalton's law of partial pressure as they state completely different things about same thing...
Raoult's law claims $p = P \cdot x$
where $p$ is the partial pressure of a gas, $P$ is the pressure of the same gas at the same temperature and volume and $x$ is the mole fraction.
Dalton's law says $p = p_\text{total} \cdot x$
where $p_\text{total}$ is the total pressure of the mixture.
You are using the symbol x to describe two different things. In Raoult's law, $p = Px$, where x is the mole fraction of the substance in the liquid phase. In Dalton's law, $p =P_{tot}y$, where y is the mole fraction of the same substance in the vapor phase. So, combining these, you have $P_{tot}y=Px$, which is really Raoult's law.
I think you are forgetting that Rault's law is used for ideal Liquid mixtures while Dalton's law is for non-reacting ideal gases. They look similar due to basic assumption of ideal interactions.
In Dalton's law partial pressures add up (PT=PA+PB), but in Rault's law since liquid has to evaporate from a surface which is approximately shared based on mole fraction their vapour pressure do not simply add up and is averaged based on mole fraction PT=xAPA+xBPB.
In Raoult's Law, $p$ is the partial pressure of the vapor in equilibrium with some liquid which has vapor pressure of $P$. $x$ is the molar fraction of this liquid (it can be mixed with other liquid or solid.
Raoult's law states that the partial vapor pressure of a substance is equal to the total vapor pressure of the solvent multiplied by the mole fraction of the substance. This also takes into account things like ion dissociation in a solution [Van't Hoff factor].
Dalton's law of partial pressures says that in a mixture of non-reacting gases, the total pressure of these gases is equal to the sum of the partial pressures of the individual gases (where the partial pressure of a given gas is calculated by multiplying the mole fraction of that gas by the total pressure).
Therefore, Raoult’s Law deals with vapor pressures while Dalton’s Law deals with non-reacting gases. However, the sum of pressures given by Raoult’s Law will results in the total amount of pressure of the [vapor] solution (which follows Dalton’s Law). So Dalton’s Law still holds in that the sum of vapor pressures is the total vapor pressure.
