How to solve a Linear Program using the Two-Phase approach? 
$$\text{Maximize }Z =  X_1 -2X_2$$
Such that:
$$3X_1 + X_2 \ge 3$$
$$2X_1 - X_2 \le 5$$
$$X_1, X_2 \ge 0$$

I've done it using CET, and found out that $\max(Z)=-6$ when $X_1=0$, $X_2=3$ which is feasible, but I really don’t know how to solve it by Two-Phase method.
Is there anyone who can help me solve this?
Really thanks about that.
Regards.
 A: To start, we have the given model:
$$\text{max } z = x_1-2x_2$$
Subject to:
$$3x_1 + x_2 \ge 3$$
$$2x_1-x_2\le5$$
$$x_1,x_2\ge0$$
We’ll need to this into standard form in standard form in respect to the Two-Phase method. However, unlike the standard form of past artificial variable methods like Big-M, we’re going to have to replace the objective function with $\text{min } w = \sum_{a_n} a_n$. Keep in mind that the first phase of the f the Two-Phase method will always replace the objective function with minimizing the sum of the artificial variables as we are trying to make them go to zero, and when we go through the simplex method of this part of the model, if we hit a point where we cannot pivot anymore and all the artificial variables are not zero, then the model is infeasible.
Thus, our model in standard form is this:
$$\text{min } w= a_1$$
Subject to:
$$3x_1 +x_2 -e_1 + a_1 =3$$
$$2x_1 -x_2 +s_2 = 5$$
From here, you’ll solve this via Simplex like you would any other model, and you should get the following tableau:
\begin{array} {|c|c|}\hline BV & w & x_1 & x_2 & e_1 & a_1 & s_2 & RHS & RT \\ \hline w & 1 & 0 & 0 & 0 & -1 & 0 & 0 & - \\ \hline x_1 & 0 & 1 & 1/3 & -1/3 & 1/3 & 0 & 1 & - \\ \hline 
s_2 & 0 & 0 & -5/3 & 2/3 & -2/3 & 1 & 3 & - \\ \hline  \end{array}
From here, we drop the objective row and the artificial variable column(s) and replace the objective row with the original standardized objective row of our original model. However, we keep the values of everything else in the tableau as such:
\begin{array} {|c|c|}\hline BV & z &
x_1 & x_2 & e_1 & s_2 & RHS & RT \\ \hline z & 1 & -1 & 2 & 0 & 0 & 0 & - \\ \hline ? & 0 & 1 & 1/3 & -1/3 & 0 & 1 & - \\ \hline s_2 & 0 & 0 & -5/3 & 2/3 & 1 & 3 & - \\ \hline  \end{array}
From this tableau, make the $x_1$ column basic and proceed like one would with any other model via the Simplex method.
