Chapter 10

Use the graphical method to find all solutions of the system of equations, correct to two decimal places. \(\left\\{\begin{aligned} x^{2}+y^{2} &=17 \\ x^{2}-2 x+y^{2} &=13 \end{aligned}\right.\)

41–44 Use a graphing calculator to graph the solution of the system of inequalities. Find the coordinates of all vertices, correct to one decimal place. $$\left\\{\begin{array}{l}{y \leq 6 x-x^{2}} \\ {x+y \geq 4}\end{array}\right.$$

Determine \(A\) and \(B\) in terms of \(a\) and \(b :\) $$\frac{a x+b}{x^{2}-1}=\frac{A}{x-1}+\frac{B}{x+1}$$

\(29-44\) Use Cramer's Rule to solve the system. $$ \left\\{\begin{aligned} x+y+z+w &=0 \\ 2 x &+w=0 \\ y-z &=0 \\ x+2 z &=1 \end{aligned}\right. $$

Write the system of equations as a matrix equation (see Example 6 ). $$ \left\\{\begin{array}{l}{2 x-5 y=7} \\ {3 x+2 y=4}\end{array}\right. $$

Find the inverse of the matrix. For what value(s) of \(x\), if any, does the matrix have no inverse? \(\left[\begin{array}{rrr}{1} & {e^{x}} & {0} \\ {e^{x}} & {-e^{2 x}} & {0} \\\ {0} & {0} & {2}\end{array}\right]\)

Find two numbers whose sum is 34 and whose difference is 10.

35–46 Solve the system of linear equations. $$\left\\{\begin{aligned} x \qquad +z+w=4 \\ y-z \qquad =-4 \\ x-2 y+3 z+w =12 \\ 2 x \qquad -2 z+5 w=-1 \end{aligned}\right.$$

Use the graphical method to find all solutions of the system of equations, correct to two decimal places. \(\left\\{\begin{array}{l}{\frac{x^{2}}{9}+\frac{y^{2}}{18}=1} \\ {y=-x^{2}+6 x-2}\end{array}\right.\)

41–44 Use a graphing calculator to graph the solution of the system of inequalities. Find the coordinates of all vertices, correct to one decimal place. $$\left\\{\begin{array}{l}{y \geq x^{3}} \\ {2 x+y \geq 0} \\ {y \leq 2 x+6}\end{array}\right.$$

Determine \(A, B, C,\) and \(D\) in terms of \(a\) and \(b :\) $$\frac{a x^{3}+b x^{2}}{\left(x^{2}+1\right)^{2}}=\frac{A x+B}{x^{2}+1}+\frac{C x+D}{\left(x^{2}+1\right)^{2}}$$

\(29-44\) Use Cramer's Rule to solve the system. $$ \left\\{\begin{array}{l}{x+y=1} \\ {y+z=2} \\ {z+w=3} \\\ {w-x=4}\end{array}\right. $$

Write the system of equations as a matrix equation (see Example 6 ). $$ \left\\{\begin{aligned} 6 x-y+z &=12 \\ 2 x+z &=7 \\ y-2 z &=4 \end{aligned}\right. $$

Find the inverse of the matrix. For what value(s) of \(x\), if any, does the matrix have no inverse? \(\left[\begin{array}{cc}{x} & {1} \\ {-x} & {\frac{1}{x-1}}\end{array}\right]\)

The sum of two numbers is twice their difference. The larger number is 6 more than twice the smaller. Find the numbers.

Use the graphical method to find all solutions of the system of equations, correct to two decimal places. \(\left\\{\begin{array}{l}{x^{2}-y^{2}=3} \\ {y=x^{2}-2 x-8}\end{array}\right.\)

Publishing Books A publishing company publishes a total of no more than 100 books every year. At least 20 of these are nonfiction, but the company always publishes at least as much fiction as nonfiction. Find a system of inequalities that describes the possible numbers of fiction and nonfiction books that the company can produce each year consistent with these policies. Graph the solution set.

For each expression, determine whether it is already a partial fraction decomposition, or whether it can be decomposed further. (a) \(\frac{x}{x^{2}+1}+\frac{1}{x+1} \quad\) (b) \(\frac{x}{(x+1)^{2}}\) (c) \(\frac{1}{x+1}+\frac{2}{(x+1)^{2}} \quad\) (d) \(\frac{x+2}{\left(x^{2}+1\right)^{2}}\)

\(45-46=\) Evaluate the determinants. $$ \left|\begin{array}{lllll}{a} & {0} & {0} & {0} & {0} \\ {0} & {b} & {0} & {0} & {0} \\ {0} & {0} & {c} & {0} & {0} \\ {0} & {0} & {0} & {d} & {0} \\\ {0} & {0} & {0} & {0} & {e}\end{array}\right| $$

Find the inverse of the matrix. For what value(s) of \(x\), if any, does the matrix have no inverse? \(\left[\begin{array}{cc}{\cos x} & {\sin x} \\ {-\sin x} & {\cos x}\end{array}\right]\)

A man has 14 coins in his pocket, all of which are dimes and quarters. If the total value of his change is \(\$ 2.75\) , how many dimes and how many quarters does he have?

35–46 Solve the system of linear equations. $$\left\\{\begin{aligned} x-y \qquad +w=0 \\ 3 x \qquad -z+2 w=0 \\ x-4 y+z+2 w =0 \end{aligned}\right.$$

Use the graphical method to find all solutions of the system of equations, correct to two decimal places. \(\left\\{\begin{array}{l}{x^{4}+16 y^{4}=32} \\ {x^{2}+2 x+y=0}\end{array}\right.\)

Furniture Manufacturing A man and his daughter manufacture unfinished tables and chairs. Each table requires 3 hours of sawing and 1 hour of assembly. Each chair requires 2 hours of sawing and 2 hours of assembly. The two of them can put in up to 12 hours of sawing and 8 hours of assembly work each day. Find a system of inequalities that describes all possible combinations of tables and chairs that they can make daily. Graph the solution set.

The following expression is a partial fraction decomposition: $$\frac{2}{x-1}+\frac{1}{(x-1)^{2}}+\frac{1}{x+1}$$ Use a common denominator to combine the terms into one fraction. Then use the techniques of this section to find its partial fraction decomposition. Did you get back the original expression?

\(45-46=\) Evaluate the determinants. $$ \left|\begin{array}{lllll}{a} & {a} & {a} & {a} & {a} \\ {0} & {a} & {a} & {a} & {a} \\ {0} & {0} & {a} & {a} & {a} \\ {0} & {0} & {0} & {a} & {a} \\\ {0} & {0} & {0} & {0} & {a}\end{array}\right| $$

Write the system of equations as a matrix equation (see Example 6 ). $$ \left\\{\begin{aligned} x-y+z &=2 \\ 4 x-2 y-z &=2 \\ x+y+5 z &=2 \\\\-x-y-z &=2 \end{aligned}\right. $$

Find the inverse of the matrix. For what value(s) of \(x\), if any, does the matrix have no inverse? \(\left[\begin{array}{ll}{\sec x} & {\tan x} \\ {\tan x} & {\sec x}\end{array}\right]\)

The admission fee at an amusement park is \(\$ 1.50\) for children and \(\$ 4.00\) for adults. On a certain day, 2200 people entered the park, and the admission fees collected totaled \(\$ 5050 .\) How many children and how many adults were admitted?

Use the graphical method to find all solutions of the system of equations, correct to two decimal places. \(\left\\{\begin{array}{l}{y=e^{x}+e^{-x}} \\ {y=5-x^{2}}\end{array}\right.\)

\(47-50=\) Solve for \(x\) $$ \left|\begin{array}{ccc}{x} & {12} & {13} \\ {0} & {x-1} & {23} \\ {0} & {0} & {x-2}\end{array}\right|=0 $$

Let $$ \begin{array}{l}{A=\left[\begin{array}{rrrr}{1} & {0} & {6} & {-1} \\ {2} & {\frac{1}{2}} & {4} & {0}\end{array}\right]} \\\ {B=\left[\begin{array}{llll}{1} & {7} & {-9} & {2}\end{array}\right]} \\ {C} & {=\left[\begin{array}{r}{1} \\ {0} \\ {-1} \\\ {-2}\end{array}\right]}\end{array} $$ Determine which of the following products are defined, and calculate the ones that are: $$ \begin{array}{ll}{A B C} & {A C B} & {B A C} \\ {B C A} & {C A B} & {C B A}\end{array} $$

A nutritionist is studying the effects of the nutrients folic acid, choline, and inositol. He has three types of food available, and each type contains the following amounts of these nutrients per ounce: (a) Find the inverse of the matrix $$\left[\begin{array}{lll}{3} & {1} & {3} \\ {4} & {2} & {4} \\ {3} & {2} & {4}\end{array}\right]$$ and use it to solve the remaining parts of this problem. (b) How many ounces of each food should the nutritionist feed his laboratory rats if he wants their daily diet to contain 10 mg of folic acid, 14 mg of choline, and 13 mg of inositol? (c) How much of each food is needed to supply 9 mg of folic acid, 12 mg of choline, and 10 mg of inositol? (d) Will any combination of these foods supply 2 mg of folic acid, 4 mg of choline, and 11 mg of inositol?

A man flies a small airplane from Fargo to Bismarck, North Dakota—a distance of 180 mi. Because he is flying into a head wind, the trip takes him 2 hours. On the way back, the wind is still blowing at the same speed, so the return trip takes only 1 h 12 min. What is his speed in still air, and how fast is the wind blowing?

Nutrition A doctor recommends that a patient take 50 mg each of niacin, riboflavin, and thiamin daily to alleviate a vitamin deficiency. In his medicine chest at home, the patient finds three brands of vitamin pills. The amounts of the relevant vitamins per pill are given in the table. How many pills of each type should he take every day to get 50 \(\mathrm{mg}\) of each vitamin? $$\begin{array}{|c|cc|} \hline {} & {\text { VitaMax }} & {\text { Vitron }} & {\text { Vitaplus }} \\ \hline \text { Niacin (mg) } & {5} & {10} & {15} \\\ {\text { Riboflavin (mg) }} & {15} & {20} & {0} \\ {\text { Thiamin (mg) }} & {10} & {10} & {10} \\ \hline\end{array}$$

A rectangle has an area of 180 \(\mathrm{cm}^{2}\) and a perimeter of 54 \(\mathrm{cm} .\) What are its dimensions?

Nutrition A cat food manufacturer uses fish and beef by-products. The fish contains 12 g of protein and 3 g of fat per ounce. The beef contains 6 g of protein and 9 g of fat per ounce. Each can of cat food must contain at least 60 g of protein and 45 g of fat. Find a system of inequalities that describes the possible number of ounces of fish and beef that can be used in each can to satisfy these minimum requirements. Graph the solution set.

\(47-50=\) Solve for \(x\) $$ \left|\begin{array}{lll}{x} & {1} & {1} \\ {1} & {1} & {x} \\ {x} & {1} & {x}\end{array}\right|=0 $$

(a) Prove that if \(A\) and \(B\) are \(2 \times 2\) matrices, then $$ (A+B)^{2}=A^{2}+A B+B A+B^{2} $$ (b) If \(A\) and \(B\) are \(2 \times 2\) matrices, is it necessarily true that $$ (A+B)^{2} \stackrel{3}{\perp} A^{2}+2 A B+B^{2} $$

A boat on a river downstream between two points, 20 mi apart, in one hour. The return trip against the current takes 2\(\frac{1}{2}\) hours. What is the boat's speed, and how fast does the current in the river flow?

Mixtures A chemist has three acid solutions at various concentrations. The first is 10\(\%\) acid, the second is 20\(\%\) , and the third is 40\(\%\) . How many milliliters of each should he use to make 100 \(\mathrm{mL}\) of 18\(\%\) solution, if he has to use four times as much of the 10\(\%\) solution as the 40\(\%\) solution?

A right triangle has an area of 84 \(\mathrm{ft}^{2}\) and a hypotenuse 25 \(\mathrm{ft}\) long. What are the lengths of its other two sides?

Shading Unwanted Regions To graph the solution of a system of inequalities, we have shaded the solution of each inequality in a different color; the solution of the system is the region where all the shaded parts overlap. Here is a different method: For each inequality, shade the region that does not satisfy the inequality. Explain why the part of the plane that is left unshaded is the solution of the system. Solve the following system by both methods. Which do you prefer? $$\left\\{\begin{aligned} x+2 y &>4 \\\\-x+y &<1 \\ x+3 y &<9 \\ x &<3 \end{aligned}\right.$$

\(47-50=\) Solve for \(x\) $$ \left|\begin{array}{lll}{1} & {0} & {x} \\ {x^{2}} & {1} & {0} \\ {x} & {0} & {1}\end{array}\right|=0 $$

Fast-Food Sales \(\quad\) A small fast-food chain with restaurants in Santa Monica, Long Beach, and Anaheim sells only hamburgers, hot dogs, and milk shakes. On a certain day, sales were distributed according to the following matrix. The price of each item is given by the following matrix.$$ \begin{array}{cc}{\text { Hamburger }} & {\text { Hot dog } \quad \text { Milk Shake }} \\ {[\$ 0.90} & {\$ 0.80} & {\$ 1.10 ]=B}\end{array} $$ $$ \begin{array}{l}{\text { (a) Calculate the product } B A \text { . }} \\\ {\text { (b) Interpret the entries in the product matrix } B A \text { . }}\end{array} $$

An encyclopedia saleswoman works for a company that offers three different grades of bindings for its encyclopedias: standard, deluxe, and leather. For each set she sells, she earns a commission based on the set’s binding grade. One week she sells one standard, one deluxe, and two leather sets and makes \(\$ 675\) in commission. The next week she sells two standard, one deluxe, and one leather set for a \(\$ 600\) commission. The third week she sells one standard, two deluxe, and one leather set, earning \(\$ 625\) in commission. (a) Let \(x, y,\) and \(z\) represent the commission she earns on standard, deluxe, and leather sets, respectively. Translate the given information into a system of equations in \(x, y,\) and \(z\). (b) Express the system of equations you found in part (a) as a matrix equation of the form \(A X=B\). (c) Find the inverse of the coefficient matrix \(A\) and use it to solve the matrix equation in part (b). How much commission does the saleswoman earn on a set of encyclopedias in each grade of binding?

A woman keeps fit by bicycling and running every day. On Monday she spends \(\frac{1}{2}\) hour at each activity, covering a total of 12\(\frac{1}{2}\) mi. On Tuesday, she runs for 12 min and cycles for 45 min, covering a total of 16 mi. Assuming her running and cycling speeds don't change from day to day, find these speeds.

The perimeter of a rectangle is 70 and its diagonal is \(25 .\) Find its length and width.

We have used the Zero-Product Property to solve algebraic equations. Matrices do not have this property. Let O represent the \(2 \times 2\) zero matrix: $$O=\left[\begin{array}{ll}{0} & {0} \\ {0} & {0}\end{array}\right]$$ Find \(2 \times 2\) matrices \(A \neq O\) and \(B \neq O\) such that \(A B=O\) . Can you find a matrix \(A \neq O\) such that \(A^{2}=O ?\)

Classroom Use A small school has 100 students who occupy three classrooms: A, B, and C. After the first period of the school day, half the students in room A move to room B, one-fifth of the students in room B move to room C, and one- third of the students in room C move to room A. Nevertheless, the total number of students in each room is the same for both periods. How many students occupy each room?