The problem I am given is max $x_1 + 3x_2$ subject to $x_1 + x_2 \leq 5, 3x_1 - x_2 \geq -3, x_1,x_2 \geq 0$
The first step I took was to put this into standard form: max $x_1 + 3x_2$ subject to $x_1 + x_2 + s_1 = 5, -3x_1 + x_2 +s_2 = 4, x_1,x_2, s_1, s_2 \geq 0$. This yields matrices $ A = \begin{bmatrix} 1 &1 &1 & 0 \\ -3 & 1 & 0 & 1 \end{bmatrix} $, $b = \begin{bmatrix} 5 \\ 3 \end{bmatrix}$, and $c = \begin{bmatrix} 1 \\ 3 \\ 0 \\ 0 \end{bmatrix}$.
Now, the way I was taught to do this is different than any example I've seen online so if you know of somewhere I could find similar examples that would be very helpful.
I now try to follow the formula given in my notes.
Let $k=1, s_1, s_2$ be basic and $x_1,x_2$ be nonbasic. then $x_B^1 = B^{-1}(b-Sx_s),B^T\pi = c_B,$ and $c_{N'} = c_{N} - A_{N}\pi$. I'm not entirely sure what all of these things should be. Mainly, $S$ and $x_s$. I think $c_N, A_N$ represent the coefficients of the nonbasic variables in the above matrices, i.e. $A_N = \begin{bmatrix} 1 &1 \\ -3 & 1 \end{bmatrix} , c_N = \begin{bmatrix} 1 \\ 3 \end{bmatrix} $. Could someone please clarify and help me work through this step and maybe the next one of $k=2$?
Thanks