I want to solve the linear program: $max. c^T x $ $\\$subject to $Ax \leq 0 \\ x\geq 0 $
If I introduce slack variables $s_i$, then I can rewrite it as $ min. -c^T x \\ s.t. A'y=0 \\ y\geq 0 , y= (x_1,..,x_n, s_1, ...,s_n)^T , A'= [A\space I_n] $
and the dual problem becomes $max -0^Tv \\ s.t. A'^Tv-\lambda +c =0 \\ \lambda \geq0$
I'm not sure if this is correct so far and also not sure how I can proceed to find a solution. The $0$ also makes this surprising for me since the value stays at $0$, how exactly will it maximise?