Relationship between the partial pressure $p_J = x_J p$ and the perfect gas law $pV = nRT$
Partial pressure is defined as the pressure a component of a gaseous mixture would exert if it alone were to occupy the container at the same temperature as that of the mixture.
If the gases in the mixture are ideal, you can apply the ideal gas equation for it $$p_A V = n_ART$$where $p_A$ and $n_A$ represents the partial pressure and number of moles of a component A. Also $$p_tV = nRT$$ where $p_t$ and $n$ represents the total pressure and number of moles of the gaseous mixture. Dividing these equations you will get $$\frac{p_A}{p_t}=x_A$$ This equation will only be true for ideal gas mixture.
Also, partial pressure is a notional concept. It is not a the actual pressure exerted by the components. It was introduced to explain and form physical properties of gaseous mixtures.