How can I represent this observation regarding options in a formula?

By observing how an option’s expiration P/L changes as its underlying asset price changes, we can discover the following system of equations:

$begin{cases}S_{Long} = C_{Long} + P_{Short} S_{Short} = P_{Long} + C_{Short} S_{Long} = -S_{Short} C_{Long} = -C_{Short} P_{Long} = -P_{Short}end{cases}$, where the $S$‘s are stocks, the $C$‘s are call options, and the $P$‘s are put options.
Linear algebra reveals that any one of these equations could be removed without loss of information.

By observing how an option’s expiration P/L changes as its strike price changes, we can discover the following similar system of equations:

$begin{cases}S_{Long} = P_{Long} + C_{Short} S_{Short} = C_{Long} + P_{Short} S_{Long} = -S_{Short} C_{Long} = -C_{Short} P_{Long} = -P_{Short}end{cases}$, again with one linear dependence shared between all five equations. It can be seen in the linked data that the call options and put options have exactly swapped behaviors, leading to the slight variation between the two systems.

How can we mathematically represent this swap and harmonize the systems? My intuition is that partial derivatives are at play, and that this may lead to a partial differential algebraic system of equations. If so, I’ll probably only understand a thoroughly curated answer, as I’m just preparing to begin studying PDEs.