Use the technique developed in this section to solve the minimization problem. $$ \begin{aligned} \text { Minimize } & C=-2 x+y \\ \text { subject to } & x+2 y \leq 6 \\ & 3 x+2 y \leq 12 \\ & x \geq 0, y \geq 0 \end{aligned} $$

Solve each linear programming problem by the method of corners. $$ \begin{array}{rr} \text { Minimize } & C=2 x+5 y \\ \text { subject to } & 4 x+y \geq 40 \\ & 2 x+y \geq 30 \\ & x+3 y \geq 30 \\ & x \geq 0, y \geq 0 \end{array} $$

The water-supply manager for a Midwest city needs to supply the city with at least 10 million gal of potable (drinkable) water per day. The supply may be drawn from the local reservoir or from a pipeline to an adjacent town. The local reservoir has a maximum daily yield of 5 million gal of potable water, and the pipeline has a maximum daily yield of 10 million gallons. By contract, the pipeline is required to supply a minimum of 6 million gallons/day. If the cost for 1 million gallons of reservoir water is $$\$ 300$$ and that for pipeline water is $$\$ 500$$, how much water should the manager get from each source to minimize daily water costs for the city? What is the minimum daily cost?

Solve each linear programming problem by the simplex method. $$ \begin{array}{ll} \text { Maximize } & P=10 x+12 y \\ \text { subject to } & x+2 y \leq 12 \\ & 3 x+2 y \leq 24 \\ & x \geq 0, y \geq 0 \end{array} $$

Solve each linear programming problem by the method of corners. $$ \begin{array}{cc} \text { Maximize } & P=3 x-4 y \\ \text { subject to } & x+3 y \leq 15 \\ & 4 x+y \leq 16 \\ & x \geq 0, y \geq 0 \end{array} $$

Determine whether the statement is true or false. If it is true, explain why it is true. If it is false, give an example to show why it is false. If at least one of the coefficients \(a_{1}, a_{2}, \ldots, a_{n}\) of the objective function \(P=a_{1} x_{1}+a_{2} x_{2}+\cdots+a_{n} x_{n}\) is positive, then \((0,0, \ldots, 0)\) cannot be the optimal solution of the standard (maximization) linear programming problem.