Chapter 6

Let \(A=\left[\begin{array}{ll}a_{11} & a_{12} \\ a_{21} & a_{22}\end{array}\right], B=\left[\begin{array}{ll}b_{11} & b_{12} \\ b_{21} & b_{22}\end{array}\right],\) and \(C=\left[\begin{array}{ll}c_{11} & c_{12} \\\ c_{21} & c_{22}\end{array}\right]\) where all the elements are real numbers. Use these matrices to show that each statement is true for \(2 \times 2\) matrices. \(A+B=B+A\) (commutative property)

Let \(A=\left[\begin{array}{ll}a_{11} & a_{12} \\ a_{21} & a_{22}\end{array}\right], B=\left[\begin{array}{ll}b_{11} & b_{12} \\ b_{21} & b_{22}\end{array}\right],\) and \(C=\left[\begin{array}{ll}c_{11} & c_{12} \\\ c_{21} & c_{22}\end{array}\right]\) where all the elements are real numbers. Use these matrices to show that each statement is true for \(2 \times 2\) matrices. \(A+(B+C)=(A+B)+C\) (associative property)

From January to June \(2012,\) Samsung and Apple spent a combined 293 million dollars on media. Apple spent 93 million dollars more than Samsung. (a) Write a system of equations whose solution gives the spending of each media company, in millions of dollars. Let \(x\) be the amount spent by Apple and \(y\) be the amount spent by Samsung. (b) Solve the system of equations. (c) Interpret the solution.

Let \(A=\left[\begin{array}{ll}a_{11} & a_{12} \\ a_{21} & a_{22}\end{array}\right], B=\left[\begin{array}{ll}b_{11} & b_{12} \\ b_{21} & b_{22}\end{array}\right],\) and \(C=\left[\begin{array}{ll}c_{11} & c_{12} \\\ c_{21} & c_{22}\end{array}\right]\) where all the elements are real numbers. Use these matrices to show that each statement is true for \(2 \times 2\) matrices. \((A B) C=A(B C)\) (associative property)

The current and estimated resident populations, \(y,\) (in percent) of Black and Spanish/Hispanic/Latino people in the United States for the years \(1990-2050\) are modeled by the following linear equations. $$\begin{aligned}&y=0.0515 x+12.3\\\&y=0.255 x+9.01\end{aligned}$$ In each case, \(x\) represents the number of years since 1990 . (Source: U.S. Census Bureau.) (a) Solve the system to find the year when these population percents were equal. (b) What percent of the U.S. resident population will be Spanish/Hispanic/Latino in the year found in part (a) (c) Graphically support the analytic solution in part (a). (d) Which population is increasing more rapidly?

Let \(A=\left[\begin{array}{ll}a_{11} & a_{12} \\ a_{21} & a_{22}\end{array}\right], B=\left[\begin{array}{ll}b_{11} & b_{12} \\ b_{21} & b_{22}\end{array}\right],\) and \(C=\left[\begin{array}{ll}c_{11} & c_{12} \\\ c_{21} & c_{22}\end{array}\right]\) where all the elements are real numbers. Use these matrices to show that each statement is true for \(2 \times 2\) matrices. Let \(A=\left[\begin{array}{ll}a_{11} & a_{12} \\ a_{21} & a_{22}\end{array}\right], B=\left[\begin{array}{ll}b_{11} & b_{12} \\ b_{21} & b_{22}\end{array}\right],\) and \(C=\left[\begin{array}{ll}c_{11} & c_{12} \\\ c_{21} & c_{22}\end{array}\right]\) where all the elements are real numbers. Use these matrices to show that each statement is true for \(2 \times 2\) matrices. \(A(B+C)=A B+A C\) (distributive property)

A wholesaler of party goods wishes to display her products at a convention of social secretaries in such a way that she gets the maximum number of inquiries about her whistles and hats. Her booth at the convention has 12 square meters of floor space to be used for display purposes. A display unit for hats requires 2 square meters, and one for whistles requires 4 square meters. Experience tells the wholesaler that she should never have more than a total of 5 units of whistles and hats on display at one time. If she receives three inquiries for each unit of hats and two inquiries for each unit of whistles on display, how many of each should she display in order to get the maximum number of inquiries? What is that maximum number?

Let \(A=\left[\begin{array}{ll}a_{11} & a_{12} \\ a_{21} & a_{22}\end{array}\right], B=\left[\begin{array}{ll}b_{11} & b_{12} \\ b_{21} & b_{22}\end{array}\right],\) and \(C=\left[\begin{array}{ll}c_{11} & c_{12} \\\ c_{21} & c_{22}\end{array}\right]\) where all the elements are real numbers. Use these matrices to show that each statement is true for \(2 \times 2\) matrices. Let \(A=\left[\begin{array}{ll}a_{11} & a_{12} \\ a_{21} & a_{22}\end{array}\right], B=\left[\begin{array}{ll}b_{11} & b_{12} \\ b_{21} & b_{22}\end{array}\right],\) and \(C=\left[\begin{array}{ll}c_{11} & c_{12} \\\ c_{21} & c_{22}\end{array}\right]\) where all the elements are real numbers. Use these matrices to show that each statement is true for \(2 \times 2\) matrices. \(c(A+B)=c A+c B,\) for any real number \(c\)

Farmer Jones raises only pigs and geese. She wants to raise no more than 16 animals, with no more than 12 geese. She spends \(\$ 50\) to raise a pig and \(\$ 20\) to raise a goose. She has \(\$ 500\) available for this purpose. Find the maximum profit she can make if she makes a profit of \(\$ 80\) per goose and \(\$ 40\) per pig. Indicate how many pigs and geese she should raise to achieve this maximum.

Let \(A=\left[\begin{array}{ll}a_{11} & a_{12} \\ a_{21} & a_{22}\end{array}\right], B=\left[\begin{array}{ll}b_{11} & b_{12} \\ b_{21} & b_{22}\end{array}\right],\) and \(C=\left[\begin{array}{ll}c_{11} & c_{12} \\\ c_{21} & c_{22}\end{array}\right]\) where all the elements are real numbers. Use these matrices to show that each statement is true for \(2 \times 2\) matrices. Let \(A=\left[\begin{array}{ll}a_{11} & a_{12} \\ a_{21} & a_{22}\end{array}\right], B=\left[\begin{array}{ll}b_{11} & b_{12} \\ b_{21} & b_{22}\end{array}\right],\) and \(C=\left[\begin{array}{ll}c_{11} & c_{12} \\\ c_{21} & c_{22}\end{array}\right]\) where all the elements are real numbers. Use these matrices to show that each statement is true for \(2 \times 2\) matrices. \((c+d) A=c A+d A,\) for any real numbers \(c\) and \(d\)

A manufacturer of refrigerators must ship at least 100 refrigerators to its two West Coast warehouses. Each warehouse holds a maximum of 100 refrigerators. Warehouse A holds 25 refrigerators already, while warehouse \(B\) has 20 on hand. It costs \(\$ 12\) to ship a refrigerator to warehouse \(\mathrm{A}\) and \(\$ 10\) to ship one to warehouse B. How many refrigerators should be shipped to each warehouse to minimize cost? What is the minimum cost?

The relationship between a professional basketball player's height \(h\) in inches and weight \(w\) in pounds was modeled by using two samples of players. The resulting equations were $$\begin{aligned}&w=7.46 h-374\\\&w=7.93 h-405\end{aligned}$$ and Assume that \(65 \leq h \leq 85\) (a) Use each equation to predict the weight to the nearest pound of a professional basketball player who is 6 feet 11 inches. (b) Determine graphically the height at which the two models give the same weight. (c) For each model, what change in weight is associated with a 1 -inch increase in height?

Let \(A=\left[\begin{array}{ll}a_{11} & a_{12} \\ a_{21} & a_{22}\end{array}\right], B=\left[\begin{array}{ll}b_{11} & b_{12} \\ b_{21} & b_{22}\end{array}\right],\) and \(C=\left[\begin{array}{ll}c_{11} & c_{12} \\\ c_{21} & c_{22}\end{array}\right]\) where all the elements are real numbers. Use these matrices to show that each statement is true for \(2 \times 2\) matrices. Let \(A=\left[\begin{array}{ll}a_{11} & a_{12} \\ a_{21} & a_{22}\end{array}\right], B=\left[\begin{array}{ll}b_{11} & b_{12} \\ b_{21} & b_{22}\end{array}\right],\) and \(C=\left[\begin{array}{ll}c_{11} & c_{12} \\\ c_{21} & c_{22}\end{array}\right]\) where all the elements are real numbers. Use these matrices to show that each statement is true for \(2 \times 2\) matrices. \((c d) A=c(d A)\)

A manufacturing process requires that oil refineries manufacture at least 2 gallons of gasoline for each gallon of fuel oil. To meet winter demand for fuel oil, at least 3 million gallons a day must be produced. The demand for gasoline is no more than 6.4 million gallons per day. If the price of gasoline is \(\$ 1.90\) per gallon and the price of fuel oil is \(\$ 1.50\) per gallon, how much of each should be produced to maximize revenue?

As the price of a product increases, businesses usually increase the quantity manufactured. However, as the price increases, consumer demand-or the quantity of the product purchased by consumers-usually decreases. The price we see in the market place occurs when the quantity supplied and the quantity demanded are equal. This price is called the equilibrium price and this demand is called the equilibrium demand. The supply of a certain product is related to its price by the equation \(p=\frac{1}{3} q,\) where \(p\) is in dollars and \(q\) is the quantity supplied in hundreds of units. (a) If this product sells for 9 dollars, what quantity will be supplied by the manufacturer? (b) Suppose that consumer demand for the same product decreases as price increases according to the equation \(p=20-\frac{1}{5} q .\) If this product sells for 9 dollars, what quantity will consumers purchase? How does this compare with the quantity being supplied by the manufacturer at this price? (c) On the basis of parts (a) and (b), what should happen to the price? Explain. (d) Determine the equilibrium price at which the quantity supplied and quantity demanded are equal. What is the demand at this price?

As the price of a product increases, businesses usually increase the quantity manufactured. However, as the price increases, consumer demand-or the quantity of the product purchased by consumers-usually decreases. The price we see in the market place occurs when the quantity supplied and the quantity demanded are equal. This price is called the equilibrium price and this demand is called the equilibrium demand. (Refer to Exercise 92 .) Suppose that supply is related to price by \(p=\frac{1}{10} q\) and that demand is related to price by \(p=15-\frac{2}{3} q,\) where \(p\) is price in dollars and \(q\) is the quantity supplied in units. (a) Determine the price at which 15 units would be supplied. Determine the price at which 15 units would be demanded. (b) Determine the equilibrium price at which the quantity supplied and quantity demanded are equal. What is the demand at this price?

Earthquake victims need medical supplies and bottled water. Each medical kit measures 1 cubic foot and weighs 10 pounds. Each container of water is also 1 cubic foot, but weighs 20 pounds. The plane can carry only \(80,000\) pounds, with total volume 6000 cubic feet. Each medical kit will aid 4 people, while each container of water will serve 10 people. How many of each should be sent in order to maximize the number of people aided? How many people will be aided?

The break-even point for a company is the point where costs equal revenues. If both cost and revenue are expressed as linear equations, the break-even point is the solution of a linear system. In each exercise, \(C\) represents cost in dollars to produce x items, and R represents revenue in dollars from the sale of \(x\) items. Use the substitution method to find the break-even point in each case-that is, the point where \(C=R .\) Then find the value of \(C\) and \(R\) at that point. $$\begin{aligned}&C=20 x+10,000\\\&R=30 x-11,000\end{aligned}$$

The break-even point for a company is the point where costs equal revenues. If both cost and revenue are expressed as linear equations, the break-even point is the solution of a linear system. In each exercise, \(C\) represents cost in dollars to produce x items, and R represents revenue in dollars from the sale of \(x\) items. Use the substitution method to find the break-even point in each case-that is, the point where \(C=R .\) Then find the value of \(C\) and \(R\) at that point. $$\begin{aligned}&C=4 x+125\\\&R=9 x-200\end{aligned}$$

Because variables appear in denominators, the system $$\begin{aligned}&\frac{5}{x}+\frac{15}{y}=16\\\&\frac{5}{x}+\frac{4}{y}=5\end{aligned}$$ is not a linear system. However, we can solve it in a manner similar to the method for solving a linear system by using a substitution-of-variable technique. Let \(t=\frac{1}{x}\) and let \(u=\frac{1}{y} \). Write a system of equations in \(t\) and \(u\) by making the appropriate substitutions.

Because variables appear in denominators, the system $$\begin{aligned}&\frac{5}{x}+\frac{15}{y}=16\\\&\frac{5}{x}+\frac{4}{y}=5\end{aligned}$$ is not a linear system. However, we can solve it in a manner similar to the method for solving a linear system by using a substitution-of-variable technique. Let \(t=\frac{1}{x}\) and let \(u=\frac{1}{y} \). Solve the system in Exercise 97 for \(t\) and \(u\).

Because variables appear in denominators, the system $$\begin{aligned}&\frac{5}{x}+\frac{15}{y}=16\\\&\frac{5}{x}+\frac{4}{y}=5\end{aligned}$$ is not a linear system. However, we can solve it in a manner similar to the method for solving a linear system by using a substitution-of-variable technique. Let \(t=\frac{1}{x}\) and let \(u=\frac{1}{y} \). Solve the given system for \(x\) and \(y\) by using the equations relating \(t\) to \(x\) and \(u\) to \(y\).

Because variables appear in denominators, the system $$\begin{aligned}&\frac{5}{x}+\frac{15}{y}=16\\\&\frac{5}{x}+\frac{4}{y}=5\end{aligned}$$ is not a linear system. However, we can solve it in a manner similar to the method for solving a linear system by using a substitution-of-variable technique. Let \(t=\frac{1}{x}\) and let \(u=\frac{1}{y} \). Refer to the first equation in the given system, and solve for \(y\) in terms of \(x\) to obtain a rational function.

Because variables appear in denominators, the system $$\begin{aligned}&\frac{5}{x}+\frac{15}{y}=16\\\&\frac{5}{x}+\frac{4}{y}=5\end{aligned}$$ is not a linear system. However, we can solve it in a manner similar to the method for solving a linear system by using a substitution-of-variable technique. Let \(t=\frac{1}{x}\) and let \(u=\frac{1}{y} \). Repeat Exercise 100 for the second equation in the given system.