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:
Airlines typically assume a value for passengers depending on gender and age. This distribution in the cabin will typically be quite random and a little variation does not do any damage.
Baggage however is weighed on the scales at check in. Each bag is normally scanned as it is passed onto the plane, so that they know where is bag is located. The values are then passed onto the pilots who are able to work out that the numbers are ok. Sometimes there are preferences to load up bags in a certain way for fuel issues, so this is very important to make sure this is still resonable.
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.