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.... You turn the problem into a one dimensional one because the
connecting cord requires coupled motion and therefore you know
that if the system moves it will move either up on the left and down on the
right or down on the left and up on the right. You then draw a big curved
arrow around the outside of the original diagram defining a positive
direction (up from the left side, over the top, down on the right--for
example.) Now a free body diagram can provide the answers. The same
trick works on the 'modified Atwood's machine.
I like to work these programs BOTH ways as it does show that there is not a
single, algorithmic way to solve all problems.