Pity, I don't know rather it helps you, but all the computer could do is addition... Minus is plus with opposite sign, multiple is many times plus and division is many times minus. That's computer in real.
Once, looking through elementary school textbook I have encountered a problem dealing with axis orientation. It was very strange, but without using complex unit this problem couldn't get proper solution.
Now you simply took from the very beginning that
P1 = F12 P2= F21
I.e. P1 = -P2
and then easily concluded
P1 + P2 = 0
Seems like it's NOT quite correct.