Use the method of this section to solve each linear programming problem. $$ \begin{array}{ll} \text { Maximize } & P=5 x+y \\ \text { subject to } & 2 x+y \leq 8 \\ & -x+y \geq 2 \\ & x \geq 0, y \geq 0 \end{array} $$

Show that the following linear programming problem $$ \begin{aligned} \text { Maximize } & P=2 x+2 y-4 z \\ \text { subject to } & 3 x+3 y-2 z \leq 100 \\ & 5 x+5 y+3 z \leq 150 \\ & x \geq 0, y \geq 0, z & \geq 0 \end{aligned} $$ has optimal solutions \(x=30, y=0, z=0, P=60\) and \(x=0, y=30, z=0, P=60 .\)

Construct the dual problem associated with the primal problem. Solve the primal problem. $$ \begin{array}{rr} \text { Minimize } & C=2 x+5 y \\ \text { subject to } & x+2 y \geq 4 \\ & 3 x+2 y \geq 6 \\ & x \geq 0, y \geq 0 \end{array} $$

You are given the final simplex tableau for the dual problem. Give the solution to the primal problem and the solution to the associated dual problem. $\begin{aligned} \text { Problem: Minimize } & C &=10 x+3 y &+10 z \\ \text { subject to } & & 2 x+y+5 z & \geq 20 \\ & & 4 x+y+z & \geq 30 \\ & x & \geq 0, y \geq 0, z & \geq 0 \end{aligned}$ $$ \begin{array}{cccccc|c} u & v & x & y & z & P & \text { Constant } \\ \hline 0 & 1 & \frac{1}{2} & -1 & 0 & 0 & 2 \\ 1 & 0 & -\frac{1}{2} & 2 & 0 & 0 & 1 \\ 0 & 0 & 2 & -9 & 1 & 0 & 3 \\ \hline 0 & 0 & 5 & 10 & 0 & 1 & 80 \end{array} $$

Rewrite each linear programming problem as a maximization problem with constraints involving inequalities of the form \(\leq\) a constant (with the exception of the inequalities \(x \geq 0, y \geq 0\), and \(z \geq 0\) ). $$ \begin{array}{ll} \text { Minimize } & C=2 x-3 y \\ \text { subject to } & 3 x+5 y \geq 20 \\ & 3 x+y \leq 16 \\ & -2 x+y \leq 1 \\ & x \geq 0, y \geq 0 \end{array} $$

Determine whether the given simplex tableau is in final form. If so, find the solution to the associated regular linear programming problem. If not, find the pivot element to be used in the next iteration of the simplex method. $$ \begin{array}{cccccc|c} x & y & z & u & v & P & \text { Constant } \\ \hline 1 & 0 & \frac{3}{5} & 0 & \frac{1}{5} & 0 & 30 \\ 0 & 1 & -\frac{19}{5} & 1 & -\frac{3}{5} & 0 & 10 \\ \hline 0 & 0 & \frac{26}{5} & 0 & 0 & 1 & 60 \end{array} $$