 Solution

The dual is $\boxed{\min b^{T}y\text{ subject to }{A}^{T}y\ge c,y\ge 0}$. Weak duality states that every primal feasible $x$ and dual feasible $y$ satisfy $c^{T}x\le b^{T}y$. Indeed,using $x\ge 0$ for the first inequality and $y\ge 0$ for the second. Thus a feasible dual value is an upper bound on every primal objective value, and a feasible pair with equal values certifies both optima.