$4\mathrm{.}3.$ The following questions below apply to the linear program be handled?

maximize z$=3$x $+13$x$2+13$x$3$

subject to

X$+$X$2<=7$ X$1+3$x$2+2$x$3<=152$x$2+3$x$1<=9$ X$1,$ X$2,$ X$320$ with optimal basis $\{$X$1,$ X$2,$ X$3\}$ and

B$-1=$

$5/2$

$-3/2$

$-3/2$

$3/2$

$-1$

$1$

$-1$

$1$

$1$

All of the questions are independent.

$($i$)$ What is the solution to the problem? What are the optimal dual variables?

$($ii$)$ What is the solution of the linear program obtained by decreasing the right$-$ hand side of the second constraint by $5?$

$($iii$)$ By how much can the right$-$hand side of the first constraint increase and de$-$ crease without changing the optimal basis?

$($iv$)$ What is the solution of the linear program obtained by increasing the coefficient of x$2$ in the objective by $15?$

$($v$)$ By how much can the objective coefficient of x$$ increase and decrease without changing the optimal basis?