91Ó°ÊÓ

In Exercises \(1-4,\) determine if the given ordered triple is a solution of the system. In Exercises \(1-4,\) determine if the given ordered triple is a solution of the system. $$ \begin{aligned} &(2,-1,3)\\\ &\left\\{\begin{array}{c} x+y+z=4 \\ x-2 y-z=1 \\ 2 x-y-2 z=-1 \end{array}\right. \end{aligned} $$

In Exercises 5–14, an objective function and a system of linear inequalities representing constraints are given. a. Graph the system of inequalities representing the constraints. b. Find the value of the objective function at each corner of the graphed region. c. Use the values in part ( \(b\) ) to determine the maximum value of the objective function and the values of \(x\) and \(y\) for which the maximum occurs. Objective Function Constraints $$\begin{aligned}&z=3 x+2 y\\\&\left\\{\begin{array}{c}x \geq 0, y \geq 0 \\\2 x+y \leq 8 \\\x+y \geq 4\end{array}\right.\end{aligned}$$

In Exercises 1–26, graph each inequality. $$x \leq 1$$

On June \(24,1948,\) the former Soviet Union blocked all land and water routes through East Germany to Berlin. A gigantic airlift was organized using American and British planes to bring food, clothing, and other supplies to the more than 2 million people in West Berlin. The cargo capacity was \(30,000\) cubic feet for an American plane and \(20,000\) cubic feet for a British plane. To break the Soviet blockade, the Western Allies had to maximize cargo capacity but were subject to the following restrictions: \(\cdot\) No more than 44 planes could be used. "The larger American planes required 16 personnel per flight, double that of the requirement for the British planes. The total number of personnel available could not exceed 512 \(\cdot\) The cost of an American flight was \(\$ 9000\) and the cost of a British flight was \(\$ 5000 .\) Total weekly costs could not exceed \(\$ 300,000\) Find the number of American and British planes that were used to maximize cargo capacity.

A theater is presenting a program for students and their parents on drinking and driving. The proceeds will be donated to a local alcohol information center. Admission is \(\$ 2.00\) for parents and \(\$ 1.00\) for students. However, the situation has two constraints: The theater can hold no more than 150 people and every two parents must bring at least one student. How many parents and students should attend to raise the maximum amount of money?

Solve each system by the method of your choice. $$\left\\{\begin{array}{l} 2 x^{2}+y^{2}-18 \\ x y-4 \end{array}\right.$$

Solve each system by the method of your choice. $$\left\\{\begin{array}{l} x^{3}+y-0 \\ x^{2}-y-0 \end{array}\right.$$

Describe a number of business ventures. For each exercise, a. Write the cost function, \(C\). b. Write the revenue function, \(R\). c. Determine the break-even point. Describe what this means. You invested \(\$ 30,000\) and started a business writing greeting cards. Supplies cost \(2 \notin\) per card and you are selling each card for \(50 \mathrm{e}\). (In solving this exercise, let \(x\) represent the number of cards produced and sold.)

In Exercises 69–70, rewrite each inequality in the system without absolute value bars. Then graph the rewritten system in rectangular coordinates. $$\left\\{\begin{array}{l} |x| \leq 1 \\ |y| \leq 2 \end{array}\right.$$

The graphs of solution sets of systems of inequalities involve finding the intersection of the solution sets of two or more inequalities. By contrast, in Exercises \(71-72,\) you will be graphing the union of the solution sets of two inequalities. Graph the union of \(y>\frac{3}{2} x-2\) and \(y<4\).