Solution
= Solution
Write $Ax=b$ as $Ax\le b$ and $-Ax\le-b$. The dual has nonnegative multipliers $u,v$ for these inequalities, with objective $b^Tu-b^Tv$ and condition $A^T(u-v)\ge c$. Every real vector $y$ can be expressed as $u-v$ with $u,v\ge0$. Hence
$$
\boxed{D':\ \min b^Ty\text{ subject to }A^Ty\ge c,\quad y\in\mathbb R^m\text{ unrestricted}.}
$$
The equality constraints remove the sign restriction on the dual variables.