Yesterday, for the Atwood's machine, the combined our two force equations to eliminate the tension, resulting in:

Mg - mg = (M + m) a

We could have obtained this right away by treating the two boxes as one combined system, and applying Newton's second law to that system.

Force responsible for the motion: Mg

Force holding the system back: mg

The net force, Mg - mg, equals the combined mass times the acceleration.