Chapter 5

In Exercises \(19-22,\) find the quadratic function \(y=a x^{2}+b x+c\) whose graph passes through the given points. $$ (-1,-4),(1,-2),(2,5) $$

Solve each system by the addition method. \(\left\\{\begin{array}{l}2 x+3 y-6 \\ 2 x-3 y-6\end{array}\right.\)

You are about to take a test that contains computation problems worth 6 points each and word problems worth 10 points each. You can do a computation problem in 2 minutes and a word problem in 4 minutes. You have 40 minutes to take the test and may answer no more than 12 problems. Assuming you answer all the problems attempted correctly, how many of each type of problem must you answer to maximize your score? What is the maximum score?

Write the partial fraction decomposition of each rational expression. $$\frac{x}{(x+1)^{2}}$$

In Exercises 1–26, graph each inequality. $$y \geq x^{2}-1$$

In Exercises \(19-22,\) find the quadratic function \(y=a x^{2}+b x+c\) whose graph passes through the given points. $$ (1,3),(3,-1),(4,0) $$

Solve each system by the addition method. \(\left\\{\begin{array}{l}3 x+2 y-14 \\ 3 x-2 y-10\end{array}\right.\)

In \(1978,\) a ruling by the Civil Aeronautics Board allowed Federal Express to purchase larger aircraft. Federal Express's options included 20 Boeing 727 s that United Airlines was retiring and/or the French-built Dassault Fanjet Falcon \(20 .\) To aid in their decision, executives at Federal Express analyzed the following data: $$\begin{array}{lcc}\hline & \text { Boeing 727 } & \text { Falcon 20 } \\\\\hline \text { Direct Operating cost } & \$ 1400 \text { per hour } & \$ 500 \text { per hour } \\\\\text { Payload } & 42,000 \text { pounds } & 6000 \text { pounds }\end{array}$$ Federal Express was faced with the following constraints: \(\cdot\) Hourly operating cost was limited to \(\$ 35,000 .\) \(\cdot\) Total payload had to be at least \(672,000\) pounds. \(\cdot\) Only twenty 727 s were available. Given the constraints, how many of each kind of aircraft should Federal Express have purchased to maximize the number of aircraft?

Write the partial fraction decomposition of each rational expression. $$\frac{x^{2}-6 x+3}{(x-2)^{3}}$$

In Exercises 1–26, graph each inequality. $$y>2^{x}$$

In Exercises \(23-24\), let \(x\) represent the first number, \(y\) the second number, and \(z\) the third number. Use the given conditions to write a system of equations. Solve the system and find the numbers. The sum of three numbers is \(16 .\) The sum of twice the first number, 3 times the second number, and 4 times the third number is \(46 .\) The difference between 5 times the first number and the second number is \(31 .\) Find the three numbers.

Solve each system by the addition method. \(\left\\{\begin{array}{l}x+2 y-2 \\ -4 x+3 y-25\end{array}\right.\)

What kinds of problems are solved using the linear programming method?

Write the partial fraction decomposition of each rational expression. $$\frac{2 x^{2}+8 x+3}{(x+1)^{3}}$$

In Exercises 1–26, graph each inequality. $$y \leq 3^{x}$$

In Exercises \(23-24\), let \(x\) represent the first number, \(y\) the second number, and \(z\) the third number. Use the given conditions to write a system of equations. Solve the system and find the numbers. The following is known about three numbers: Three times the first number plus the second number plus twice the third number is \(5 .\) If 3 times the second number is subtracted from the sum of the first number and 3 times the third number, the result is \(2 .\) If the third number is subtracted from 2 times the first number and 3 times the second number, the result is \(1 .\) Find the numbers.

Solve each system by the addition method. \(\left\\{\begin{array}{l}2 x-7 y-2 \\ 3 x+y--20\end{array}\right.\)

What is an objective function in a linear programming problem?

Write the partial fraction decomposition of each rational expression. $$\frac{x^{2}+2 x+7}{x(x-1)^{2}}$$

In Exercises 1–26, graph each inequality. $$y \geq \log _{2}(x+1)$$

Solve each system in Exercises \(25-26\) $$ \left\\{\begin{array}{l} \frac{x+2}{6}-\frac{y+4}{3}+\frac{z}{2}=0 \\ \frac{x+1}{2}+\frac{y-1}{2}-\frac{z}{4}=\frac{9}{2} \\ \frac{x-5}{4}+\frac{y+1}{3}+\frac{z-2}{2}=\frac{19}{4} \end{array}\right. $$

Solve each system by the addition method. \(\left\\{\begin{array}{l}4 x+3 y-15 \\ 2 x-5 y-1\end{array}\right.\)

What is a constraint in a linear programming problem? How is a constraint represented?

Write the partial fraction decomposition of each rational expression. $$\frac{3 x^{2}+49}{x(x+7)^{2}}$$

In Exercises 1–26, graph each inequality. $$y \geq \log _{3}(x-1)$$

Solve each system in Exercises \(25-26\) $$ \left\\{\begin{array}{l} \frac{x+3}{2}-\frac{y-1}{2}+\frac{z+2}{4}=\frac{3}{2} \\ \frac{x-5}{2}+\frac{y+1}{3}-\frac{z}{4}=-\frac{25}{6} \\ \frac{x-3}{4}-\frac{y+1}{2}+\frac{z-3}{2}=-\frac{5}{2} \end{array}\right. $$

Solve each system by the addition method. \(\left\\{\begin{array}{l}3 x-7 y-13 \\ 6 x+5 y-7\end{array}\right.\)

In your own words, describe how to solve a linear programming problem.

Write the partial fraction decomposition of each rational expression. $$\frac{x^{2}}{(x-1)^{2}(x+1)}$$

In Exercises 27–62, graph the solution set of each system of inequalities or indicate that the system has no solution. $$\left\\{\begin{array}{l} 3 x+6 y \leq 6 \\ 2 x+y \leq 8 \end{array}\right.$$

Solve each system by the addition method. \(\left\\{\begin{array}{l}3 x-4 y-11 \\ 2 x+3 y--4\end{array}\right.\)

Describe a situation in your life in which you would really like to maximize something, but you are limited by at least two constraints. Can linear programming be used in this situation? Explain your answer.

Write the partial fraction decomposition of each rational expression. $$\frac{x^{2}}{(x-1)^{2}(x+1)^{2}}$$

In Exercises 27–62, graph the solution set of each system of inequalities or indicate that the system has no solution. $$\left\\{\begin{array}{l} x-y \geq 4 \\ x+y \leq 6 \end{array}\right.$$

Solve each system by the addition method. \(\left\\{\begin{array}{l}2 x+3 y--16 \\ 5 x-10 y-30\end{array}\right.\)

In Exercises 29-32, determine whether each statement makes sense or does not make sense, and explain your reasoning. In order to solve a linear programming problem, I use the graph representing the constraints and the graph of the objective function.

Write the partial fraction decomposition of each rational expression. $$\frac{5 x^{2}-6 x+7}{(x-1)\left(x^{2}+1\right)}$$

In Exercises 27–62, graph the solution set of each system of inequalities or indicate that the system has no solution. $$\left\\{\begin{array}{l} 2 x-5 y \leq 10 \\ 3 x-2 y>6 \end{array}\right.$$

In Exercises \(29-30,\) solve each system for \((x, y, z)\) in terms of the nonzero constants \(a, b,\) and \(c\) $$ \left\\{\begin{array}{c} a x-b y-2 c z=21 \\ a x+b y+c z=0 \\ 2 a x-b y+c z=14 \end{array}\right. $$

Solve each system by the method of your choice. $$\left\\{\begin{array}{l} 3 x^{2}+4 y^{2}-16 \\ 2 x^{2}-3 y^{2}-5 \end{array}\right.$$

Solve each system by the addition method. \(\left\\{\begin{array}{l}3 x-4 y+1 \\ 3 y-1-4 x\end{array}\right.\)

Write the partial fraction decomposition of each rational expression. $$\frac{5 x^{2}-9 x+19}{(x-4)\left(x^{2}+5\right)}$$

In Exercises 27–62, graph the solution set of each system of inequalities or indicate that the system has no solution. $$\left\\{\begin{array}{l} 2 x-y \leq 4 \\ 3 x+2 y>-6 \end{array}\right.$$

In Exercises \(29-30,\) solve each system for \((x, y, z)\) in terms of the nonzero constants \(a, b,\) and \(c\) $$ \left\\{\begin{array}{c} a x-b y+2 c z=-4 \\ a x+3 b y-c z=1 \\ 2 a x+b y+3 c z=2 \end{array}\right. $$

Solve each system by the method of your choice. $$\left\\{\begin{array}{l} x+y^{2}-4 \\ x^{2}+y^{2}-16 \end{array}\right.$$

Solve each system by the addition method. \(\left\\{\begin{array}{l}5 x-6 y+40 \\ 2 y-8-3 x\end{array}\right.\)

In Exercises 29-32, determine whether each statement makes sense or does not make sense, and explain your reasoning. I need to be able to graph systems of linear inequalities in order to solve linear programming problems.

Write the partial fraction decomposition of each rational expression. $$\frac{5 x^{2}+6 x+3}{(x+1)\left(x^{2}+2 x+2\right)}$$

In Exercises 27–62, graph the solution set of each system of inequalities or indicate that the system has no solution. $$\left\\{\begin{array}{l} y>2 x-3 \\ y<-x+6 \end{array}\right.$$

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