I'm aware of some aircraft that have systems to calculate weight and balance. For those aircraft that don't have such systems, how does the pilot know they are within weight and balance limits?
While I'm not a pilot, I think it goes along these lines:
These values are then plugged into computers or worked out manually on a paper loadsheet to make sure they are good and within limits.
In fact the weight systems you are referring to are quite unusual I think, see this question.