Section 7.1
92. Investing her bonus. Donna invested her $33,000 bonus and received a total of $970 in interest after one year. If part of the money returned 4% and the remainder 2.25%, then how much did she invest at each rate? x+y=33000 .04x+2.25y=970 x=-y+33000 .04-y+33000+2.25y=970
-4y+1320+.0225y=970
-.0175y=-1320+970
-.0175y-.0175=-350-.0175
y=20,000 x+20,000=33,000 x=33,000-20,000 x=13000 100. Ticket sales. Tickets for a concert were sold to adults for $3 and to students for $2. If the total receipts were $824 and twice as many adult tickets as student tickets were sold, then how many of each were sold?

3x+2y=824 x=2y 32y+2y=824
6y+2y=824
8y=824 y=103 x=103*2 x=206 104. Bonus and taxes. A company has an income of $100,000 before paying taxes and a bonus. The bonus B is to be 20% of the income after deducting income taxes T but before deducting the bonus. So
B = 0.20(100,000 - T).
Because the bonus is a deductible expense, the amount of income tax T at a 40% rate is 40% of the income after deducting the bonus. So
T = 0.40(100,000 - B).

a) Use the accompanying graph to estimate the values of T and B that satisfy both equations.
(35, 15) b) Solve the system algebraically to find the bonus and the amount of tax.
B = .2(100,000-.4(100000 – B))
B = .2(100000 – 40000 + .4B)
B = 20000 – 8000 + .08B
.92B = 12000
B = $13042.48
T = .4(100000 – 13042.48)
T = .4(86957.52)
T = $34783

106. Free market. The equations S = 5000 + 200x and D = 9500 - 100x express the supply S and the demand D, respectively, for a popular compact disc brand in terms of its price x (in dollars).

a) Graph the equations on the same coordinate system.
S = 200x + 5000
S = 200(0) + 5000
S = 5000
S = 200(5) + 5000
S = 6000
D = -100x + 9500
D = -100(0) + 9500
D = 9500
D = -100(5) + 9500
D = -500 + 9500
D = 9000

b) What happens to the supply as the price increases?
Increases
c) What happens to the demand as the price increases?
Decreases
d) The price at which supply and demand are equal is called the equilibrium price. What is the equilibrium price? $15.00 * *
Section 7.2:

52. 2y – x = 3 x = 3y – 5
2y – (3y -5) = 3
2y -3y + 5 = 3
-y + 5 = 3
-y = -2 y = 2 (Add y value to original equation) check 2y – x = 3
2y – x = 3 2(2) -1= 3
2(2) – x = 3 4 – 1 = 3 (true)
4 – x = 3
-x = -1 x = 1
Answer: (1, 2)

72. Books and magazines. At Gwen’s garage sale, all books were one price, and all magazines were another price. Harriet bought four books and three magazines for $1.45, and June bought two books and five magazines for $1.25. What was the price of a book and what was the price of a magazine?
4b + 3m =$1.45 4b + 3m =$1.45
2b +5m = $1.25 4b + 3(.15) =$1.45 4b + .45 =$1.45
1(4b + 3m =$1.45) 4b = $1.00
-2(2b +5m = $1.25) b = .25

4b + 3m =$1.45 2b +5m = $1.25
-4b -10m =-$2.50 2b +5(.15) = $1.25 -7m = -1.05 2b + .75 = $1.25 m = $.15 2b = .50 b =.25

check: 4b + 3m =$1.45 4(.25) + 3(.15) =$1.45 1.00 + .45 = $1.45 (true)
Answer: All garage sale books cost .25 each and all magazines cost .15 each. * * *
78. Low-fat yogurt. Ziggy’s Famous Yogurt blends regular yogurt that is 3% fat with its no-fat yogurt to obtain lowfat yogurt that is 1% fat. How many pounds of regular yogurt and how many pounds of no-fat yogurt should be mixed to obtain 60 pounds of low-fat yogurt?

* x = pounds of regular yogurt (3% fat) * y = pounds of no fat yogurt (0% fat) * 60 = pounds of low fat yogurt (1% fat) * * 0.03 * x + 0.00 * y = 0.01 * 60 * x + y =60 * * y = 60 – x * * 0.03 * x = 0.01 * 60 * 0.03x0.03 = 0.01*600.03 * x = 20 * 20 pounds of regular yogurt is needed. * * y = 60 – x * y = 60 – 20 = 40 * 40 pounds of no fat yogurt is needed. * * 20 pounds of regular yogurt needs to be mixed with 40 pounds of no fat yogurt to get 60 pounds of low fat yogurt.

82. Super Bowl contender. The probability that San Francisco plays in the next Super Bowl is nine times the probability that they do not play in the next Super Bowl. The probability that San Francisco plays in the next Super Bowl plus the probability that they do not play is 1. What is the probability that San Francisco plays in the next Super Bowl?

x = Probability that SF plays in the next Super Bowl y = Probability that SF does not play in the next Super Bowl which equals 9x

9x + x = 1
10x = 1 x = 110 = 10% probability of SF not playing in the next Super Bowl y = 9x = 9 * 110 = 910 = 90% probability of SF playing in the next Super Bowl