Applications of 2 x 2 Systems of Equations
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1 Systems of equations can be applied to any problem situation which satisfies two criteria: Two unknowns can be identified and assigned variables. Two equations can be formulated using the variables. 1. Theresa has a total of 40 DVDs in two categories. The number of her action DVDs is 10 more than twice the number of her documentary DVDs. Determine a system of equations that can be used to find how many action DVDs, a, and documentary DVDs, d, Theresa has. a. What are the two variables and what does each represent? b. What two equations can be formulated from the information in the problem using the variables? c. Solve the system of equations to find the number of action and documentary DVDs Theresa has. What is the most appropriate method to solve this system? d. Is the solution reasonable to the problem situation? Explain. 2. Let x and y represent the measures of a pair of supplementary angles. One angle is 36 degrees less than twice the second angle. a. What two equations can be formulated from the problem situation? b. Solve the system of equations to find the measures of the two supplementary angles. 2012, TESCCC 09/21/12 page 1 of 5
2 Two other types of problem situations that can be solved using systems of equations are the rate and mixture problems. Rate Problems Set up two equations in the y = mx + b form. Manually Solve by substitution. Calculator Solve by graphing or table. 3. Green Grass, Inc. charges a flat rate of $15 per hour for lawn services. Their main competitor S & G Lawn Care charges an initial fee of $30 plus $10 per hour. a. Complete the table below. Green Grass, Inc. S & G Lawn Care Hours Process Cost Process Cost x y starting or base amount + rate x b. Use the pattern in the table to write an algebraic expression to represent each company. c. Extend the chart to find when the charges for each company are equal. Verify this by finding the solution to the system of equations graphically and algebraically. d. Explain how to determine when it would be better to use Green Grass, Inc. and when it would be better to use their competitor, S & G Lawn Care. 2012, TESCCC 09/21/12 page 2 of 5
3 Mixture Problems Set up two equations in standard form: ax by c Manually Solve by elimination. Calculator Solve by matrices. 4. Felix has a bug collection consisting of beetles and spiders. He has 75 bugs in all. Altogether they have 526 legs. a. Identify the variables. b. Write an equation representing total bugs. c. Write an equation representing total legs. d. Solve the system to find the number of beetles and the number of spiders in the collection. 2012, TESCCC 09/21/12 page 3 of 5
4 Guided Practice 1. Sweets for the Sweet sells candy in large bins. Each bin normally contains one type of candy, but a customer dumped the gumballs and mega jawbreakers into the same bin. The sales clerk checks the sales records and knows the bin must contain 847 pieces of candy. He weighs the bin and finds it weighs grams. An individual gumball weighs 2.5 grams. An individual jawbreaker weighs 4.2 grams. How many of each type of candy are in the mixed up bin? a. Identify the variables. b. Set up two equations to represent this situation. c. Solve the system to find the number of gumballs and jawbreakers. Justify the solution in terms of the problem situation. 2. Alexander and his brother Albert work at two different department stores. Alexander is paid a base salary of $250 per week plus a 30% commission on all sales. Albert is paid a base salary of only $125 per week, but gets a 40% commission on all sales. a. Identify the independent and dependent variables. b. Write an equation to represent Alexander s weekly pay as a function of amount of sales. c. Write an equation to represent Albert s weekly pay as a function of amount of sales. d. Calculate the amount of sales each must make to have equal weekly salaries. What will that salary be? 2012, TESCCC 09/21/12 page 4 of 5
5 Practice Problems Complete problems on your own paper. Show all work. 1. Coach Barker bought supper for all the participating volleyball players after the tournament. The food was ordered from Fast Food s to Go, and the choices were hamburger meals for $2.50 or hot dog meals for $1.75. Coach Barker ordered a total of 45 meals and the ticket, not including tax, was $ How many of each meal did she order? 2. At a local bookstore, John purchased a history textbook and a magazine that cost a total of $32, not including tax. If the price of the history textbook, h, is $4 more than 3 times the price of the magazine, m, which system of linear equations could be used to determine the price of the history textbook and the magazine? 3. Gouda cheese and Edam cheese cost different amounts per kilogram. A gift box containing 3 kg of Gouda and 2 kg of Edam costs $ Another gift box containing 4 kg of Gouda and 5 kg of Edam costs $ Assuming the total cost only reflects the cost of the cheeses, what is the price of each cheese per kilogram? 4. Green Grass, Inc. charges an initial fee of $10 plus $12 per hour for lawn services. Their main competitor S & G Lawn Care charges an initial fee of $25 plus $8 per hour. a. Complete a table. b. Write an algebraic expression to represent each company. c. Find the solution to the system of equations graphically and algebraically. d. Explain how to determine when it would be better to use Green Grass, Inc. and when it would be better to use their competitor, S & G Lawn Care One ship sailed from Hawaii on a course represented by the equation y x 215. Another 2 ship left Tahiti on a course represented by the equation y 7x a. Determine a viewing window to show the intersection of the equations representing the paths of the two ships. Give appropriate domain and range values for this viewing window. b. Draw a graphic representation of the two ships. c. Determine the point at which the path of the ships would cross. 6. After a baseball game was over the turnstile showed that 1787 people attended. The total cash received for tickets was $5792. Reserved seats cost $4.00 each and general admission cost $3.00 each. Compute the number of each type of ticket sold for the game. 2012, TESCCC 09/21/12 page 5 of 5
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