1. Solve the following systems of linear equations.

(a) x - 2y = 4    x/3 - y = 4/3	Ans:
(b) x - 2y = 4   -4x + 8y = -16	Ans:
(c) 2x/3 - y = 2     x + y/2 = 1	Ans:
(d) 0.2x - 0.4y = 0.6   -0.8x + 1.6y = -2.4	Ans:

2. A factory makes both basketballs and soccer balls. Each basketball requires 2 minutes on the forming machine, while each soccer ball requires 1 minute. Each basketball requires 1 minute on the inflating machine, while each soccer ball requires 1.5 minutes. Assuming that these two machines run constantly, what is the total number of basketballs and soccer balls they can make in an hour?

basketballs   soccer balls  

     

3. Your portfolio manager has suggested the following two stocks for investment purposes: Beta Banking, Inc. and Gamma Go Corp. Beta Banking stock costs $36 per share, while Gamma Go stock costs $45 per share. You have $6,300 to invest and wish to hold twice as many Gamma Go shares as Beta Banking shares. How many units of each stock should you buy?

units of Beta Banking   units of Gamma Go  

     

4. Total box office revenues for "The Cable Guy" in the first two weeks after its release totaled $45 million, but had dropped by 50% in the second week. What were the box office revenues for each of the first two weeks? ^*

* Source: Columbia/The New York Times, July 1, 1996, p. D7.

$ million in week 1   $ million in week 2  

     

5. My personal trainer, Big Erik, wants me to take 900 mg. of taurine (an amino acid) and 10 mg vanadium (a mineral) per day to prepare me for the Mr/Ms Hofstra competition. I have just gone down to the health food store and purchased two supplements: Vanadyl Pro-Gro, containing 225 mg of taurine and 2 mg vanadium per tablet, and Muscle XS, containing 150 mg taurine and 2 mg vanadium per tablet. How many of each tablet should I take per day in order to obtain the exact recommended dosage.

mg. of taurine Vanadyl Pro-Gro tablets

and

Muscle XS tablets mg. of vanadium

6. Solve the following systems of linear equations.

(a) x - y + z = 4       2x - y = 6       x + y - z = 6	Ans:
(b) x - y + z = 4       2x - y = 6       x + y - 3z = 0	Ans:
(c) 0.4x    + 0.6z = 1.0       x/2 +y/3 + z/6 = 1       x + y - z = 1	Ans:
(d) x + y + 5z + w = 1       y + 2z + w = 0       x + 3y + 7z + 2w = 1       x + y + 5z + 2w = 1	Ans:

7. Your nutritionist, Jeff Mongillo, has decided that you need 350 mg. of Vitamin C, 4,200 I.U. of Vitamin A, and 400 I.U. of Vitamin D per day^Ý, and recommends the following supplements: Megadose Supra, containing 50 mg. of Vitamin C, 1,000 I.U. of Vitamin A, and 100 I.U. of Vitamin D per capsule, Crazy Caps, containing 100 mg. of Vitamin C, 200 I.U. of Vitamin A, and 100 I.U. of Vitamin D per capsule, and AC D-Free's, containing 50 mg. of Vitamin C and 500 I.U. of Vitamin A per capsule, but no vitamin D. How many of each should you take to obtain exactly the recommended daily dosages?

Megadose Supras   Crazy Caps   D-Free ACs  

   

Ý The current (1999) US Recommended Daily allowances for these supplements are: Vitamin A: 5,000 I.U., Vitamin C: 60 mg., and Vitamin D: 400 I.U. Thus Healthy Harry's recommendation is a little low on Vitamin A, generous on Vitamin C---many nutritionists feel that the Vitamin C RDA should at least be doubled---and dead right on Vitamin D.

8. The top three U.S. eyeglass retailers in 1994 were Cole Vision, Lens Crafters and Pearle Vision. Together, they accounted for 66% of total market sales. The combined share of Cole Vision and Lens Crafters was twice the share of Pearle Vision, and Lens Crafters' share was 2% less than the combined share of Cole Vision and Pearle Vision.^Ý What was the market share of each of the three companies?

Cole Vision:%   Lens Crafters:%   Pearle Vision:%  

   

Ý "Total market share" is based on total sales by the top ten retailers. All figures were rounded to the nearest percentage point. Source: 20/20 Optical Group Data Base/The New York Times, April 1, 1995, p. L35.

9. The college jogging team goes through jogging shoes like water. The coach usually orders three brands of jogging shoes which they obtain at cost: Gauss Jordans, Roebecks, and K Scottish. Gauss Jordans cost the team $50 per pair, Roebecks $50 and K Scottish $45. One year, the team went through a total of 120 pairs at a total cost of $5,700. Given that the team went through as many pairs of Gauss Jordans as Roebecks, how many pairs of each brand of jogging shoes did they use?

pairs Gauss Jordans   pairs Roebecks   pairs K Scottish  

   

10. In an attempt to measure traffic flow near Ellipse Park, town officials in Wayville placed traffic counters on two portions of Eastbound Street, as shown in the figure, and measured 100 cars per hour on both streets. (Arrows indicate the direction of traffic flow. Note that Up Avenue is a two-way street.)

(a) Is there sufficient information to measure the traffic flow around Round St.? Explain your answer.
(b)Write down the general solution of the associated system of linear equations.
(c) Can southbound traffic along Up Avenue exceed northbound traffic? Explain your answer by referring to the solution in part (b).

11. The following system of one-way streets appears in Enormous State University's Fraternity Row. (The boxed numbers give the number of vehicles per hour.)

(a) Find the general solution to a system of linear equations one would use to calculate the hourly traffic flow along Rush St., Pledge Ave, Fasttrack Ave. and Depledge Blvd.
(b) If there is no traffic along Fasttrack Ave, what is the lightest possible traffic along Rush Street?
(c) Given that 50 cars per hour go down Depledge Blvd., what is the heaviest traffic down Fasttrtack?
(d) Specify a particular solution in which there are 1,000 cars down Depledge Blvd.

12. You run a surf fashion store, D'Amico Surf, in South Park Mall. Two days ago you were notified of an upcoming vacancy in another part of the Mall, and you are seriously considering relocating D'Amico Surf there, as you suspect that there is more traffic in that area. Unfortunately, you can't spare one of your staff to count the total traffic going past the locations, but you do have partial information, based on data provided by the South Park Mall management, as shown in the following diagram.

(The arrows indicate the net direction of shopper traffic. The numbers in boxes indicate the number of shoppers per minute during peak shopping hours.)

You then get to work:

(a) You first set up a system of linear equations with 5 unknowns, and solve it.
(b) You then calculate the quantity (New Location Traffic)-(Present Location Traffic) What do you find?