# Two- phase simplex method problem

I have this problem:

Minimize:

$x_1+3x_2-x_3$

Subject to:

\begin{align} 2x_1+x_2+3x_3 \geq 3\\ -x_1+x_2\geq1\\ -x_1-5x_2+x_3\leq4\\ x_1,x_2,x_3, \geq 0\\ \end{align}

I need help solving it though the two phase simplex method. I keep getting stuck and need help determining whether or not it has a feasible region and what the feasible points are. I think I keep cycling the pivots but am not sure.

It might be wrong but I think we cannot minimize $\,x_1+3x_2-x_3\,$
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Let $\,x_1=0,\ x_2\geq1,\ x_3=4x_2$, $\$then we have $$2x_1+x_2+3x_3=13x_2>3$$ $$-x_1+x_2=x_2\geq1$$ $$-x_1-5x_2+x_3=-x_2<4$$ $$x_1=0,\ \ x_2>0,\ \ x_3=4x_2>0$$ Now all inequalities hold, so we can make $\,x_2\,$be infinitely large, and that will make $$\,x_1+3x_2-x_3=-x_2\,$$ be infinitely small.
As a result, we cannot minimize $$\,x_1+3x_2-x_3\,$$ because there is no such a minimum value.