Consider the following linear programming problem. Min $3A+5B$

$s.t.$

$,1A+4B21$

$,2A+1B7$

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

$-2A+6B0$

$,A,B0$

$($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 $7A+3B.$ Find the optimal solution and the value of the objective function.

Min $3$A $+5$B

s$.$t$.$

$1$A $+4$B

$2$A $+1$B $>=7$

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

$2$A $+6$B $>=0$

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

surplus for

$2$A $+1$B $>=7$

slack for

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

surplus for

$2$A $+6$B $>=0$

$($c$)$

Suppose the objective function is changed to max

$7$A $+3$B$.$

Find the optimal solution and the value of the objective function.

at $($A$,$ B$)=$