We can easily see that $1=2=3=1$ is a feasible point for the following linear

program:$=41+22+33$

Subject to:

$21+32+3<=12$

$1+42+23<=10$

$31+2+3<=10$

$1,2,3>=0$a$)$ Write this LP with slack variables. What are the values of the slack variables at the point

$1=2=3=1?$ What is the value of the objective function at this point?b$)$ Without solving the problem, identify which variables are basic and which are non$-$basic

at this point.

c$)$ What is the dual of this LP$?$ Write it in minimum form with all surplus variables.

d$)$ Note that each constraint in the LP has a unique variable, and that there is a dual

variable that corresponds to that constraint. These are called complimentary variables.

List all the pairs of complimentary variables in your primal and dual LP$.$

e$)$ Find any feasible point for the dual. What are the values of all the dual variables

$($including slack variables$)$ at this point? What is the value of the dual objective function

at this point?

f$)$ Multiply together the values of all the pairs of complimentary variables $($giving $6$

products$)$ then add them together. What do you get? Compare this to the difference

between the value of the dual objective function you calculated in $($e$)$ and the value of

the primal objective function from $($a$).$ What do you notice?

g$)$ Solve both the primal and the dual LPs $($you can use a solver$).$ What are the values of all

the variables at the optimum? What do you notice about the product of each of the

pairs of complimentary variables? This is a property of primal$/$dual linear programs

called Tucker Duality.

$4.[5$ points$]$ Solve the following linear progr