$4.$ Consider the following linear programming problem: Minimize z$=10$ x$\_1+$x$\_2$ subject to

$[4$ x$\_1+$x$\_2<=32$; $2$ x$\_1+$x$\_2>=12$; $2$ x$\_1-$x$\_2<=4$; $-2$ x$\_1+$x$\_2<=8$; x$\_1,$ x$\_2>=0]$

$($a$)[5$ marks$]$ By introducing three slack variables and one surplus variable write this problem in canonical form. $($b$)[20$ marks$]$ Now, introducing one artificial variable, use the two$-$phase simplex method to solve the problem.