Is the Linear Program below unfeasible, unlimited or does it have an optimal solution? Properly justify your answer, making use of the dual program if it makes sense – if it doesn't, justify why
\begin{align} max_{x_i}-x_1 + 5x_2 + 2x_3 - 7x_4 - x_5\\ x_2 + x_3 - x_4 \geq 13\\ x_1 - x_2 + 2x_4 + 2x_5 \leq 4\\ x_2,x_4,x_5 \geq 0 x_3 \geq -2\end{align}
I don't think there is a single viable solution, but I don't know how to continue. This is what i was able to do.
Primal \begin{align} max_{x_i}-x_1 + 5x_2 + 2x_3 - 7x_4 - x_5\\ x_2 + x_3 - x_4 \geq 13\\ -x_1 + x_2 - 2x_4 - 2x_5 \geq -4\\ x_2,x_4,x_5 \geq 0 x_3 \geq -2 \end{align}
Dual \begin{align} min_{y_1,y_2}13y_1 - 4y_2 \\ 0y_1 + y_2 \leq -1\\ y_1 + y_2 \leq 5\\ y_1 +0y_2 \leq 2\\ y_1 +2y_2 \leq -7\\ 0y_1 +2y_2 \leq -1\end{align}
$A = \begin{pmatrix}0&1&1&1&0\\-1&1&0&2&2\\ \end{pmatrix}$
$b = (13,-4)^T$
$x = (x_1,x_2,x_3,x_4,x_5)^T$
$y = (y_1,y_2)^T$
Thanks for the help.