Chapter 5

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. $$\begin{aligned} &x^{2}+y^{2}=13\\\ &x^{2}-y^{2}=5 \end{aligned}$$

In Exercises \(19-30,\) solve each system by the addition method. \(x+y=6\) \(x-y=-2\)

Write the partial fraction decomposition of each rational expression. $$\frac{2 x^{2}-18 x-12}{x^{3}-4 x}$$

Graph each inequality. $$ y \geq x^{2}-1 $$

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 supply 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. \(\cdot\) 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.

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. $$\begin{aligned} &4 x^{2}-y^{2}=4\\\ &4 x^{2}+y^{2}=4 \end{aligned}$$

In Exercises \(19-30,\) solve each system by the addition method. \(2 x+3 y=6\) \(2 x-3 y=6\)

Write the partial fraction decomposition of each rational expression. $$\frac{6 x-11}{(x-1)^{2}}$$

Graph each inequality. $$ y>2^{x} $$

A theater is presenting a program on drinking and driving for students and their parents. 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?

Find the quadratic function \(y=a x^{2}+b x+c\) whose graph passes through the given points. $$(-1,6),(1,4),(2,9)$$

Solve each system by the addition method. $$\begin{aligned} x^{2}-4 y^{2} &=-7 \\ 3 x^{2}+y^{2} &=31 \end{aligned}$$

In Exercises \(19-30,\) solve each system by the addition method. \(3 x+2 y=14\) \(3 x-2 y=10\)

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

Graph each inequality. $$ y \leq 3^{x} $$

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 do to maximize your score? What is the maximum score?

Find the quadratic function \(y=a x^{2}+b x+c\) whose graph passes through the given points. $$(-2,7),(1,-2),(2,3)$$

In Exercises \(19-30,\) solve each system by the addition method. \(\begin{aligned} x+2 y &=2 \\\\-4 x+3 y &=25 \end{aligned}\)

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

Graph the solution set of each system of inequalities or indicate that the system has no solution. $$ \begin{aligned}&3 x+6 y \leq 6\\\&2 x+y \leq 8\end{aligned} $$

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. $$\begin{aligned} &3 x^{2}+4 y^{2}-16=0\\\ &2 x^{2}-3 y^{2}-5=0 \end{aligned}$$

In Exercises \(19-30,\) solve each system by the addition method. \(2 x-7 y=2\) \(3 x+y=-20\)

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

Graph the solution set of each system of inequalities or indicate that the system has no solution. $$ \begin{aligned}&x-y \geq 4\\\&x+y \leq 6\end{aligned} $$

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

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. $$\begin{aligned} 16 x^{2}-4 y^{2}-72 &=0 \\ x^{2}-y^{2}-3 &=0 \end{aligned}$$

In Exercises \(19-30,\) solve each system by the addition method. \(4 x+3 y=15\) \(2 x-5 y=1\)

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

Graph the solution set of each system of inequalities or indicate that the system has no solution. $$ \begin{aligned}&2 x-5 y \leq 10\\\&3 x-2 y>6\end{aligned} $$

What is an objective function in a linear programming problem?

Solve each system by the addition method. $$\begin{aligned} &x^{2}+y^{2}=25\\\ &(x-8)^{2}+y^{2}=41 \end{aligned}$$

In Exercises \(19-30,\) solve each system by the addition method. \(3 x-7 y=13\) \(6 x+5 y=7\)

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

Graph the solution set of each system of inequalities or indicate that the system has no solution. $$ \begin{aligned}&2 x-y \leq 4\\\&3 x+2 y>-6\end{aligned} $$

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

In Exercises \(19-30,\) solve each system by the addition method. \(3 x-4 y=11\) \(2 x+3 y=-4\)

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

Graph the solution set of each system of inequalities or indicate that the system has no solution. $$ \begin{aligned}&y>2 x-3\\\&y<-x+6\end{aligned} $$

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

You throw a ball straight up from a rooftop. The ball misses the rooftop on its way down and eventually strikes the ground. A mathematical model can be used to describe the relationship for the ball's height above the ground, \(y,\) after \(x\) seconds. Consider the following data: $$\begin{array}{|c|c|}\hline \begin{array}{c}x, \text { seconds after the } \\\\\text { ball is thrown }\end{array} & \begin{array}{c}y, \text { ball's height, in feet, } \\\\\text { above the ground }\end{array} \\ \hline 1 & 224 \\\\\hline 3 & 176 \\\\\hline 4 & 104 \\\\\hline\end{array}$$ a. Find the quadratic function \(y=a x^{2}+b x+c\) whose graph passes through the given points. b. Use the function in part (a) to find the value for \(y\) when \(x=5 .\) Describe what this means.

Solve each system by the addition method. $$\begin{aligned} &y^{2}-x=4\\\ &x^{2}+y^{2}=4 \end{aligned}$$

In Exercises \(19-30,\) solve each system by the addition method. \(2 x+3 y=-16\) \(5 x-10 y=30\)

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

Graph the solution set of each system of inequalities or indicate that the system has no solution. $$ \begin{aligned}&y<-2 x+4\\\&y

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.

A mathematical model can be used to describe the relationship between the number of feet a car travels once the brakes are applied, \(y,\) and the number of seconds the car is in motion after the brakes are applied, \(x .\) A research firm collects the following data: $$\begin{array}{cc}\begin{array}{c}x, \text { seconds in motion } \\\\\text { after brakes are applied } \end{array} & \begin{array}{c}y, \text { feet car travels } \\\\\text { once the brakes are applied }\end{array} \\\\\hline 1 & 46 \\\\\hline 2 & 84 \\\\\hline 3 & 114\end{array}$$ a. Find the quadratic function \(y=a x^{2}+b x+c\) whose graph passes through the given points. b. Use the function in part (a) to find the value for \(y\) when \(x=6 .\) Describe what this means.