Chapter 5

In Exercises \(5-18,\) solve each system by the substitution method. $$ \left\\{\begin{array}{l} y=-\frac{1}{2} x+2 \\ y=\frac{3}{4} x+7 \end{array}\right. $$

Solve each system. $$\left\\{\begin{aligned} 7 z-3 &=2(x-3 y) \\ 5 y+3 z-7 &=4 x \\ 4+5 z &=3(2 x-y) \end{aligned}\right.$$

write the partial fraction decomposition of each rational expression. $$ \frac{4 x^{2}-7 x-3}{x^{3}-x} $$

Graph each inequality. $$y

In Exercises \(19-28,\) solve each system by the addition method. $$ \left\\{\begin{array}{l} x^{2}+y^{2}=13 \\ x^{2}-y^{2}=5 \end{array}\right. $$

In Exercises \(19-30,\) solve each system by the addition method. \(\left\\{\begin{array}{l}x+y=1 \\ x-y=3\end{array}\right.\)

In Exercises 19-22, find the quadratic function \(y=a x^{2}+b x+c\) whose graph passes through the given points. $$(-1,6),(1,4),(2,9)$$

Use the two steps for solving a linear programming problem. 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: \- 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 .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.

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

Graph each inequality. $$y

In Exercises \(19-28,\) solve each system by the addition method. $$ \left\\{\begin{array}{l} 4 x^{2}-y^{2}=4 \\ 4 x^{2}+y^{2}=4 \end{array}\right. $$

In Exercises \(19-30,\) solve each system by the addition method. $$ \left\\{\begin{array}{l} x+y=1 \\ x-y=3 \end{array}\right. $$

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

Use the two steps for solving a linear programming problem. 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 dollar for parents and 1.00 dollar 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?

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

Graph each inequality. $$y \geq x^{2}-9$$

In Exercises \(19-28,\) solve each system by the addition method. $$ \left\\{\begin{array}{l} x^{2}-4 y^{2}=-7 \\ 3 x^{2}+y^{2}=31 \end{array}\right. $$

In Exercises \(19-30,\) solve each system by the addition method. $$ \left\\{\begin{array}{l} 2 x+3 y=6 \\ 2 x-3 y=6 \end{array}\right. $$

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

Use the two steps for solving a linear programming problem. 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. $$ \frac{x}{(x+1)^{2}} $$

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

In Exercises \(19-28,\) solve each system by the addition method. $$ \left\\{\begin{array}{l} 3 x^{2}-2 y^{2}=-5 \\ 2 x^{2}-y^{2}=-2 \end{array}\right. $$

In Exercises \(19-30,\) solve each system by the addition method. $$ \left\\{\begin{array}{l} 3 x+2 y=14 \\ 3 x-2 y=10 \end{array}\right. $$

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

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

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

In Exercises \(19-28,\) solve each system by the addition method. $$ \left\\{\begin{array}{l} 3 x^{2}+4 y^{2}-16=0 \\ 2 x^{2}-3 y^{2}-5=0 \end{array}\right. $$

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

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.

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

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

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

In Exercises \(19-28,\) solve each system by the addition method. $$ \left\\{\begin{aligned} 16 x^{2}-4 y^{2}-72 &=0 \\ x^{2}-y^{2}-3 &=0 \end{aligned}\right. $$

In Exercises \(19-30,\) solve each system by the addition method. $$ \left\\{\begin{array}{l} 2 x-7 y=2 \\ 3 x+y=-20 \end{array}\right. $$

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.

What is an objective function in a linear programming problem?

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

Graph each inequality. $$y \geq \log _{2}(x+1)$$

In Exercises \(19-28,\) solve each system by the addition method. $$ \left\\{\begin{array}{l} x^{2}+y^{2}=25 \\ (x-8)^{2}+y^{2}=41 \end{array}\right. $$

In Exercises \(19-30,\) solve each system by the addition method. $$ \left\\{\begin{array}{l} 4 x+3 y=15 \\ 2 x-5 y=1 \end{array}\right. $$

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

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

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

Graph each inequality. $$y \geq \log _{3}(x-1)$$

In Exercises \(19-28,\) solve each system by the addition method. $$ \left\\{\begin{array}{l} x^{2}+y^{2}=5 \\ x^{2}+(y-8)^{2}=41 \end{array}\right. $$

In Exercises \(19-30,\) solve each system by the addition method. $$ \left\\{\begin{array}{l} 3 x-7 y=13 \\ 6 x+5 y=7 \end{array}\right. $$

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

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

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