Consider the following linear programming problem.

Min $4A+5B$

$s.t.$

$,1A+4B21$

$,2A+1B7$

$,3A+1\mathrm{.}5B21$

$,-2A+6B0$

$,A,B0$

$($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 $1A+4B21$

surplus for $2A+1B7$

slack for $3A+1\mathrm{.}5B21$

surplus for $-2A+6B0$

$($c$)$ Suppose the objective function is changed to max $8A+3B.$ Find the optimal solution and the value of the objective function.

at $(A,B)=$