Q$1.$ Consider the following linear model:

maxZ$=2x_1+8x_2+x_3$

$s.t.2x_1+4x_{2}12$

$2x_1+2x_2+x_{3}4$

$x_1,x_{2}0$

$($a$)$ What is the Basic Feasible Solution with basic variables $(x_1,s_1)?$

$($b$)$ Solve above LP with Simplex Method.

Q$2.$ We will use the same linear model in Q$1.$

$($a$)$ Write the dual form of primal model.

$($b$)$ Solve the dual form with graphical solution approach.

$($c$)$ Compare the objective value obtained from Q$1-($b$)$ and Q$2-($b$).$ Are they same or not?