A system transitions from an initial state p_1, V_1 to a final...

According to the first law of thermodynamics, the heat supplied to the system is given by $\Delta Q = \Delta U + \Delta W$, where $\Delta U$ is the change in internal energy and $\Delta W$ is the work done by the system.

Rearranging the equation gives $\Delta U = \Delta Q - \Delta W$.

Internal energy is a state function, meaning its change $\Delta U$ depends only on the initial and final states of the system and is independent of the path taken.

Since the initial state $p_1, V_1$ and the final state $p_2, V_2$ are the same for both methods, the change in internal energy $\Delta U$ must be identical for both methods.

Therefore, the quantity $\Delta Q - \Delta W$ must be identical for both methods.