I'm trying to understand what exactly the complementary slackness conditions for a primal dual pair is and how it's calculated.
I understand that we have a given linear programming problem. And that said problem has a dual problem (min becomes max ext.) It seems to me like it has something to do with the difference between the min and max solutions but aren't they supposed to be the same?
I suspect you are talking here about $cx\le by$. Typically for feasible (but non-optimal) solutions for the primal and dual pair we have an inequality here and this becomes an equality when both are optimal.