Consider the linear program:

Maximize z$=9$ x$\_2+$x$\_3-2$ x$\_5-$x$\_6.$

subject to:

$[5$ x$\_2+50$ x$\_3+$x$\_4+$x$\_5=10,$; x$\_1-15$ x$\_2+2$ x$\_3=2,$; x$\_2+$x$\_3+$x$\_5+$x$\_6=6,$; x$\_$j$>=0($j$=1,2,...,6)]$

a$)$ Find an initial basic feasible solution, specify values of the decision variables, and tell which are basic. b$)$ Transform the system of equations to the canonical form for carrying out the simplex routine. c$)$ Is your initial basic feasible solution optimal? Why? d$)$ How would you select a column in which to pivot in carrying out the simplex algorithm? e$)$ Having chosen a pivot column, now select a row in which to pivot and describe the selection rule. How does this rule guarantee that the new basic solution is feasible? Is it possible that no row meets the criterion of your rule? If this happens, what does this indicate about the original problem?