 Solution

With primal constraints $Ax=b$, the dual variable is unrestricted in sign:Membership of $x$ in the nonnegative orthant still yields $c^{T}x\le y^{T}Ax$, but the next step is now the equality $y^{T}Ax=y^{T}b$, so no sign condition on $y$ is needed. Equivalently an equality constraint can be replaced by two opposite inequalities whose nonnegative multipliers combine into one free multiplier.