Solution
= Solution
Feasibility makes both sums $x^T(A^Ty-c)$ and $y^T(b-Ax)$ nonnegative. Their sum is the gap $b^Ty-c^Tx$. At optimal points the assumed equal optimal values make that gap zero, forcing each nonnegative sum to vanish. Hence
$$
\boxed{(y^TA-c^T)x=0,\qquad y^T(Ax-b)=0.}
$$
In fact every individual nonnegative product in the two sums vanishes, giving the componentwise <complementary slackness> conditions too.