Transcript
Ch.6-7 Systems of Linear Equations and Inequalities
Example ? Solve a system of linear equations for 2 variables.
When a popular playwright opened a new play, he had to decide whether to open the show on Broadway or off Broadway. For example, he decided to open his last play off Broadway. From information provided by his producer, the following equations were developed:
43,500x ? y = 1,295,000
27,000x ? y = 440,000
where x represents the number of weeks that the show ran and y represents the profit or loss from the show. The first equation is for Broadway, and the second equation is for off Broadway. (Lial, p. 275 6.1.19)
a. Solve this system of equations to determine when the profit or loss from the show will be equal for each venue.
b. What is the amount of that profit or loss?
c. Graph both equations. Discuss which venue is favorable for the show.
018415000
On the basis of data from 2000 to 2010, the median weekly earnings y in year x for men and women is approximated by the equations given below, where x = 0 corresponds to 2000 and y is in constant 2010 dollars. (Lial, p. 276 6.1.23)
Men: 10y – 13x = 8110
Women: 2y – 9x = 1248
If these equations remain valid in the future, when will the median income of men and women be the same?
Example? Set up and solve a system of linear equations for 2 variables.
Shirley Cicero has $16,000 invested in Boeing and GE. The Boeing stock currently sells for $30 a share and the GE stock sells for $70 a share. If the GE stock triples in value and the Boeing stock goes up 50%, her stock will be worth $34,500. How many shares of each stock does she own? (Lial, p. 276 6.1.29)
lefttopNote: there is an error in this typed solution. In the first line (2), it should read 34,500 and NOT 345,000
A 200-seat theatre has tickets for sale at $8 for adults and $5 for children. If all the seats were filled and the total ticket income was $1435, how many adults and how many children were in the audience?
(Lial, p. 276 6.1.25)
Example ? Set up a system of linear equations for more than 2 variables.
A person plans to invest a total of $100,000 in a money market account, a bond fund, an international stock fund, and a domestic stock fund. She wants 60% of her investment to be conservative (money market and bonds). She wants the amount in international stocks to be one-fourth of the amount in domestic stocks. Finally, she needs an annual return of $4,000. Assuming she gets annual returns of 2.5% on the money market account, 3.5% on the bond fund, 5% on the international stock fund, and 6% on the domestic stock fund. Set up the system of equations describing the given information.
(Lial, p.290 6.3.2)
The U-Drive Rent-A-Truck company plans to spend $5 million on 200 new vehicles. Each van will cost $20,000, each small truck $25,000, and each large truck $35,000. Past experience shows that they need twice as many vans as small trucks. Set up the system of equations describing the given information. Solve the system. (Lial, p. 294 6.3.1)
Jennifer took clothes to the cleaners three times last month. First, she brought 3 shirts and 1 pair of slacks and paid $10.96. Then she brought 7 shirts, 2 pairs of slacks, and a sports coat and paid $30.40. Finally, she brought 4 shirts and 1 sports coat and paid $14.45. How much was she charged for each shirt, each pair of slacks, and each sports coat? (Lial, p. 6.3.5)
lefttopNOTE: Or see your class notes for the substitution method.
Example ? Set up a system of linear inequalities and graph the feasible region.
A college student works in both the school cafeteria and library. She works no more than 12 hours per week at the cafeteria, and no more than 14 hours per week at the library. She must work at least 20 hours each week. (Lial, 7.1.53 in MyMathLab)
Let x =number of hours worked at the cafeteria per week and y =number of hours worked at the library per week. Write a system of inequalities that describes all the given conditions.
Graph the feasible region for the system of inequalities.
251460142875
c) If the student earns $12 and $10 per hour in the cafeteria and library respectively, what assignment of hours will maximize her salary?
A company produces handmade shawls and afghans. They spin the yarn, dye it, and then weave it. A shawl requires 1 hour of spinning, 1 hour of dyeing, and 1 hour of weaving. An afghan requires 2 hours of spinning, 1 hour of dyeing, and 4 hours of weaving. They spend at most 8 hours each day for spinning, 6 hours for dyeing, and 14 hours for weaving. (Lial, p. 344 7.1.51)
Summarize the data as a system of inequalities. (You might want to set up a table to help you)
Then graph the feasible region of this system of inequalities.
Example ? Set up optimization problems
A bicycle manufacturer builds one-, three- and ten-speed models. The bicycles need both aluminum and steel. The company has available 91,800 units of steel and 42,000 units of aluminum. The one-, three-, and ten-speed models need, respectively, 20, 30 and 40 units of steel and 12, 21, and 16 units of aluminum. The company makes $8 per one-speed bike, $12 per three-speed, and $24 per ten-speed. Set up the optimization problem (optimize profit with respect to production levels). (Lial, p. 376 7.5.5)
The Fashion Store has $8000 available each month for advertising. Newspaper ads cost $400 each and no more than 20 can be run per month. Radio ads cost $200 each and no more than 30 can run per month. TV ads cost $1200 each, with a maximum of 6 available each month. Approximately 2000 women will see each newspaper ad, 1200 will hear each radio ad, and 10,000 will see each TV ad. Set up the optimization problem to maximize exposure. (Lial, p. 376 7.5.11)
