Question

How is the synthesis of all of atmospheric variable necessary to make accurate predictions about changing weather?

Answer 1

Weather is complex but there are only really 2 variables in the atmosphere.

Explanation:

I already know this is probably not the answer you want. Unfortunately this is what is actually going on in the atmosphere.

There are only 2 ways the atmosphere can change and all other variables are controlled by this 2 ways. These are temperature and absolute humidity.

The best way to analyze why this is the case is to look at the most common variable that people think of the atmosphere, the pressure.
Atmospheric pressure is entirely dependent on temperature and water content. The warmer an air mass is the higher the pressure. The drier an airmass is the higher the pressure.