$(40$ points$)$

Consider the following LP

maxz$=3x_1+4x_2+x_3$

$s.t.x_1+x_2+x_{3}50$

$2x_1-x_2+x_{3}15$

$x_1+x_2=10$

$x_1,x_2,x_{3}0$

$($a$)$ Find the dual of this LP

$($b$)$ Find optimal solution for the above primal LP and its dual using Excel Solver or Python Gurobi.

$($c$)$ Find the range of values of the objective function coefficient of $x_l$ and the right$-$hand side of the $1^{st}$ constraint for which the current basis remains optimal.

$($d$)$ What is the relationship between the optimal solution of LP and its dual.