How to solve this linear program? I was given the following linear program with the supposed answers to be $x_1 = 45/103$, $x_2 = 27/103$, $x_3 = 31/103$. Howerver, I tried to solve it using the Simplex Algorithm with no success, online LP solvers also couldn't seem to find the solutions, but checking the answers work with the LP. Could someone please show me how to solve this?
$min$ $z,$ $subject$ $to$
$-2x_1 + x_2 + 3x_3 \leq  z$
$x_1 - 4x_2 + 3x_3 \leq z$
$3x_1 + 3x_2 - 6x_3 \leq z$
$x_1 + x_2 + x_3 = 1$
$x_1,x_2,x_3  \geq 0$
I converted the above to
$Minimize$ $p = z$ $subject$ $to$
$z + 2x_1 - x_2 - 3x_3 \geq 0$
$z - x_1 + 4x_2 - 3x_3  \geq 0$
$z - 3x_1 - 3x_2 + 6x_3  \geq 0$
$x_1 + x_2 + x_3 = 1$  

I know this is old, but you could only add Z as a new variable inside your simplex formulation and let:
X1 = Z
and just change all the other variables you had as:
Xn = Xn+1 So you would have this system of equations:
Min P = X1
S.T:
X1 +2X2 -X3 -3X4 >= 0
X1 -X2 +4X3 -3X4 >= 0
X1 -3X2 -3X3 +6X4 >= 0
X2 + X3 +X4 = 1
Solve using Two Phase method or Big M method (You could multiply all >= 0 constraints by -1 to simplify the problem) and then you just change back your variables:
Z = X1
Xn = Xn-1
Solving this problem gets you Z = 2/7 if I recall.
