$(35$ points$)$ Consider the following problem:

min,$3x_1+2x_2+7x_3$

$s.t.,-x_1+x_2=10$

$,2x_1-x_2+x_{3}10$

$,x_1,x_2,x_{3}0$

$($a$)$ Construct the dual of the given linear program.

$($b$)$ Suppose the shadow price for the first functional constraint of the primal LP is $7$;

that for the second functional primal constraint is $5.$ Use complementary slackness

to find an optimal solution for the primal problem.

$($c$)$ What is the range of feasibility for the right$-$hand$-$side coefficient of the second

primal functional constraint?

$($d$)$ Suppose the coefficients associated with $x_1$ are simultaneously changed to $c_1=$

$2,a_{11}=-2,a_{21}=3.$ Without resolving the problem, determine whether the current

basis is still optimal.