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
$Minimize$ $p = z$ $subject$ $to$
$z + 2x_1 - x_2 - 3x_3 \geq 0$
$z - x_1 + 4x_2 - 3x_3  \geq 0$
$z - 3x_1 - 3x_2 + 6x_3  \geq 0$
$x_1 + x_2 + x_3 = 1$  

Multiply the first equation by $45$, second by $27$, third by $31$ and add them up:
$103z\geq 30x_1+30x_2+30x_3=30$
Hence $z_{min}={30\over103}$
After that just solve the system of equations letting each of the three equations equal to $z_{min}$.
