Consider the following tableau for a standard linear programming problem:

$\left[\begin{array}{ccccccc}x& y& z& s_1& s_2& M& ?\\ 1& p& 0& 6& 0& 0& 53\\ 0& q& 1& -2& 1& 0& 13\\ 0& r& 5& 4& 0& 1& -17\end{array}\right].(We$ want $to$ maximize $M).$

$($a$)$ Find the basic solution that corresponds to this simplex tableau.

$$ For each of the following questions $($separately$),$ give an example of values of $p,q$ and $r$ to make the statement true:

$($b$)$ The basic solution for this tableau is an optimal solution.

$($c$)$ The basic solution for this tableau is not optimal, and feasible values of $M$ can be positive numbers of arbitrarily large magnitude. $($i$.$e$.,$ max$=).$