Let $A \in \mathbb R^{m \times n }, b \in \mathbb R^m$ and $0 \neq c \in \mathbb R^m$. Prove: Either the system $Ax = c$ or the system $A^T y = 0, c^T y = 1$ has a solution.
Looks a bit like Farkas Lemma but the $c^T y = 1$ confuses me. Can someone give me a hint?
I know two versions of Farkas Lemma:
$$ \text{Either} \quad Ax = b, x \geq 0 \quad \text{or} \quad A^T y \geq 0, b^T y < 0 \quad \text{is solvable.}$$
$$ \text{Either} \quad Ax \leq b \quad \text{or} \quad A^T y = 0, y \geq 0, b^T y < 0 \quad \text{is solvable.}$$
Hint: To get a nonnegative variable into the mix, replace a free variable with a difference of nonnegative variables.