PROBLEM II $($give detailed explanations and avoid using computer tools for this problem, this is for you to gain more intuition on how to solve linear programs$)$

A certain linear program has the following as its dual:

Maximize $15$u $+22$v $+30$w

such that $2$u $+3$v $+5$w $<=60$

u $+$ v $+$ w $<=20$

u $+$ v $<=10$

u$,$ v$,$ w are nonnegative

Find the optimal solution to the primal linear program using the following steps:

$($a$)$ Use the simplex method to determine the values in the final optimal tableau for this given dual problem. Give the first four iterations of the simplex method and explain each iteration on how goes to find the optimal values.

$($b$)$ Using the optimal tableau of the dual problem, what are the basic variables in the optimal solution to the primal linear program and what is the optimal value?

$($c$)$ If we change the second constraint$$s right side to $21($in the given dual program$),$ what is the change that will happen to the optimal solution to the primal program?

$($d$)$ At optimality, how much slack is there? If none, is the first constraint in the primal linear program binding in the optimal solution $($meaning the slack variable in the first constraint is zero at optimality$)?$