The radioactive isotope ^23592U finally changes to the stable isotope ^20782Pb through a series of α and β^- emissions. If x is the number of α…

Only $\alpha$ emission changes the mass number, and each $\alpha$ lowers it by $4$ (and the atomic number by $2$). The mass number falls from $235$ to $207$, a drop of $28$, so $x=\dfrac{28}{4}=7$. The $7$ alpha particles lower the atomic number by $14$, to $92-14=78$. Each $\beta^-$ emission raises the atomic number by $1$ (a neutron changes to a proton) without changing the mass number, so reaching $Z=82$ needs $y=82-78=4$ beta particles. Check on charge: $92-2(7)+4=82$, as required for lead, and the mass check $235-4(7)=207$ also holds. Therefore $x-y=7-4=3$.