problem $3$

Consider the following linear program.

Min $8x+12Y$

$s.t.1x+3Y9$

$,2x+2Y10$

$,6x+2Y18$

$x,Y0,$

$-$Use the graphical solution procedure to find the optimal solution. Make sure you label each axis and the origin. Identify the feasible area. Identify the optimal solution. Place variable $x$ on the horizontal axis.

$-$What is the optimal solution?

$-$What is the value of the objective function?

$-$Assume the objective function for $x$ changes from $8$ to $6.$

Graph the new objective function on the graph. Does the optimal solution change?

$-$Assume the objective function for $x$ remains $8,$ but the coefficient for $Y$ changes from $12$ to $6.$ Graph the new objective function on the graph. Does the optimal solution change?