Intuitive Explanation of an Atwood Machine

[BlackHayze]

So I'm working through a physics a textbook on my own, and because of that don't always have a professor who can help explain concepts I don't understand, so I'm turning to you good folks.

Could someone explain why both masses need to be taken into account when determining the acceleration of one side of an Atwood Machine? And maybe the way I started doing it led me to have a harder time, but I basically found the tension by subtracting the smaller weight from the heavier, then dividing by the total weight got me the correct answer.

I'm fairly certain this way of going about it is wrong, but most of what I see online about the correct way isn't making intuitive sense to me. Could anyone try to explain to me how it works and why the math works out the way it does? Thanks in advance!

 

[Simon Bridge]

The core concept is that the weights are pulling in opposite directions.
The formal approach is to isolate the masses, draw free body diagrams, and apply Newton's Laws to each separately.