91Ó°ÊÓ

Solving a System In Exercises \(35-40,\) use any method to solve the system.$$\left\\{\begin{array}{c}{-x+3 y=17} \\ {4 x+3 y=7}\end{array}\right.$$

Optimal Revenue An accounting firm has 780 hours of staff time and 272 hours of reviewing time available each week. The firm charges \(\$ 1600\) for an audit and \(\$ 250\) for a tax return. Each audit requires 60 hours of staff time and 16 hours of review time. Each tax return requires 10 hours of staff time and 4 hours of review time. What numbers of audits and tax returns will yield an optimal revenue? What is the optimal revenue?

Solving a System of Linear Equations In Exercises \(25 - 46\) , solve the system of linear equations and check any solutions algebraically. $$ \left\\{ \begin{array} { r r } { x + 2 y - 7 z = } & { - 4 } \\ { 2 x + y + z = } & { 13 } \\ { 3 x + 9 y - 36 z = } & { - 33 } \end{array} \right. $$

In Exercises 33-46, sketch the graph (and label the vertices) of the solution set of the system of inequalities. $$\left\\{\begin{array}{l}{x-y^{2}>0} \\ {x-y>2}\end{array}\right.$$

A fruit grower raises crops \(A\) and \(B\) . The yield is 300 bushels per acre for crop A and 500 bushels per acre for crop B. Research and available resources indicate the following constraints. \(\cdot\) The fruit grower has 150 acres of land for raising the crops. \(\cdot\) It takes 1 day to trim an acre of crop \(A\) and 2 days to trim an acre of crop \(B\) , and there are 240 days per year available for trimming. \(\cdot\) It takes 0.3 day to pick an acre of crop \(A\) and 0.1 day to pick an acre of crop \(B,\) and there are 30 days per year available for picking. What is the optimal acreage for each fruit? What is the optimal yield?

Media Selection A company has budgeted a maximum of \(\$ 1,000,000\) for national advertising of an allergy medication. Each minute of television time costs \(\$ 100,000\) and each one-page newspaper ad costs \(\$ 20,000 .\) Each television ad is expected to be viewed by 20 million viewers, and each newspaper ad is expected to be seen by 5 million readers. The company's market research department recommends that at most 80\(\%\) of the advertising budget be spent on television ads. What is the optimal amount that should be spent on each type of ad? What is the optimal total audience?

Acid Mixture Thirty liters of a 40\(\%\) acid solution is obtained by mixing a 25\(\%\) solution with a 50\(\%\) solution. (a) Write a system of equations in which one equation represents the amount of final mixture required and the other represents the percent of acid in the final mixture. Let \(x\) and \(y\) represent the amounts of the 25\(\%\) and 50\(\%\) solutions, respectively. (b) Use a graphing utility to graph the two equations in part (a) in the same viewing window. As the amount of the 25\(\%\) solution increases, how does the amount of the 50\(\%\) solution change? (c) How much of each solution is required to obtain the specified concentration of the final mixture?

Fuel Mixture Five hundred gallons of 89 -octane gasoline is obtained by mixing 87 -octane gasoline with \(92-\) octane gasoline. (a) Write a system of equations in which one equation represents the amount of final mixture required and the other represents the amounts of \(87-\) and 92 -octane gasolines in the final mixture. Let \(x\) and \(y\) represent the numbers of gallons of 87 -octane and 92 -octane gasolines, respectively. (b) Use a graphing utility to graph the two equations in part (a) in the same viewing window. As the amount of 87 -octane gasoline increases, how does the amount of 92 -octane gasoline change? (c) How much of each type of gasoline is required to obtain the 500 gallons of 89 -octane gasoline?

Investment Portfolio A total of \(\$ 32,000\) is invested in two municipal bonds that pay 5.75\(\%\) and 6.25\(\%\) simple interest. The investor wants an annual interest income of \(\$ 1900\) from the investments. What amount should be invested in the 5.75\(\%\) bond?

In Exercises 53-60, write a system of inequalities to describe the region. Rectangle: vertices at \((4,3),(9,3),(9,9),(4,9)\)