Consider the following linear programming problem.

Min $6$A $+7$B

s$.$t$.$

$1$A $+4$B $<=22$

$2$A $+1$B $>=9$

$3$A $+1\mathrm{.}5$B $<=24$

$2$A $+6$B $>=2$

A$,$ B $>=0$

$($a$)$

Find the optimal solution using the graphical solution procedure and the value of the objective function. $\_\_\_\_\_\_$ at $($A$,$ B$)=\_\_\_\_\_\_$

$($b$)$

Determine the amount of slack or surplus for each constraint.

slack for

$1$A $+4$B $<=22$

surplus for

$2$A $+1$B $>=9$

slack for

$3$A $+1\mathrm{.}5$B $<=24$

surplus for

$2$A $+6$B $>=2$

$($c$)$

Suppose the objective function is changed to max

$8$A $+3$B$.$

Find the optimal solution and the value of the objective function. $\_\_\_\_$ at $($A$,$ B$)=\_\_\_\_\_$