Use the method of this section to solve each linear programming problem. $$ \begin{aligned} \text { Minimize } & C=x+2 y \\ \text { subject to } & 4 x+7 y \leq 70 \\ & 2 x+y=20 \\ & x \geq 0, y \geq 0 \end{aligned} $$

Steinwelt Piano manufactures uprights and consoles in two plants, plant I and plant II. The output of plant I is at most \(300 /\) month, whereas the output of plant II is at most \(250 /\) month. These pianos are shipped to three warehouses that serve as distribution centers for the company. To fill current and projected orders, warehouse A requires a minimum of 200 pianos/month, warehouse B requires at least 150 pianos/month, and warehouse \(\mathrm{C}\) requires at least 200 pianos/month. The shipping cost of each piano from plant I to warehouse A, warehouse \(\mathrm{B}\), and warehouse \(\mathrm{C}\) is $\$ 60, \$ 60\(, and \)\$ 80$, respectively, and the shipping cost of each piano from plant II to warehouse A, warehouse B, and warehouse \(\mathrm{C}\) is $\$ 80, \$ 70\(, and \)\$ 50$, respectively. Use the method of this section to determine the shipping schedule that will enable Steinwelt to meet the warehouses' requirements while keeping the shipping costs to a minimum.

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 \(z \geq 0\) ). $$ \begin{array}{ll} \text { Maximize } & P=2 x+y-2 z \\ \text { subject to } & x+2 y+z \geq 10 \\ & 3 x+4 y+2 z \geq 5 \\ & 2 x+5 y+12 z \leq 20 \\ & x \geq 0, y \geq 0, z \geq 0 \end{array} $$

Solve each linear programming problem by the simplex method. $$ \begin{array}{cc} \text { Maximize } & P=3 x+4 y+5 z \\ \text { subject to } & x+y+z \leq 8 \\ & 3 x+2 y+4 z \leq 24 \\ x & \geq 0, y \geq 0, z \geq 0 \end{array} $$

Solve each linear programming problem by the simplex method. $$ \begin{array}{lr} \text { Maximize } & P=x+4 y-2 z \\ \text { subject to } & 3 x+y-z \leq 80 \\ & 2 x+y-z \leq 40 \\ & -x+y+z \leq 80 \\ x & \geq 0, y \geq 0, z & \geq 0 \end{array} $$

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