Algebra II - Chapter 3 Review

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1 Algebra II - Chapter 3 Review Multiple Choice Identify the choice that best completes the statement or answers the question. 1. An independent system of two linear equations has an infinite number of solutions. a. always b. sometimes c. never 2. Equivalent systems of two linear equations have the same solutions. a. always b. sometimes c. never 3. A system of two linear inequalities has a solution. a. always b. sometimes c. never Short Answer Solve the system by graphing A rental car agency charges a flat fee of $32.00 plus $3.00 per day to rent a certain car. Another agency charges a fee of $30.50 plus $3.25 per day to rent the same car. a. Write a system of equations to represent the cost c for renting a car at each agency for d days. b. Using a graphing calculator, find the number of days for which the costs are the same. Round your answer to the nearest whole day. Without graphing, classify each system as independent, dependent, or inconsistent. 7. Solve the system by the method of substitution. 8.

2 9. A group of 52 people attended a ball game. There were three times as many children as adults in the group. Set up a system of equations that represents the numbers of adults and children who attended the game and solve the system to find the number of children who were in the group. Use the elimination method to solve the system Solve the system of inequalities by graphing Your club is baking vanilla and chocolate cakes for a bake sale. They need at most 25 cakes. You cannot have more than 10 chocolate cakes. Write and graph a system of inequalities to model this system. 17. An exam consists of two parts, Section X and Section Y. There can be a maximum of 90 questions. There must be at least 5 more questions in Section Y than in Section X. Write a system of inequalities to model the number of questions in each of the two sections. Then solve the system by graphing. 18. Find the values of x and y that maximize the objective function P = 3x + 2y for the graph. What is the maximum value?

3 19. Given the system of constraints, name all vertices. Then find the maximum value of the given objective function. Maximum for 20. Your computer supply store sells two types of inkjet printers. The first, type A, costs $137 and you make a $50 profit on each one. The second, type B, costs $100 and you make a $0 profit on each one. You can order no more than 100 printers this month, and you need to make at least $00 profit on them. If you must order at least one of each type of printer, how many of each type of printer should you order if you want to minimize your cost? 21. Graph the system of constraints. Then find the values of x and y that maximize. Essay 22. A fish market buys tuna for $.50 per pound and spends $1.50 per pound to clean and package it. Salmon costs $2.00 per pound to buy and $2.00 per pound to clean and package. The market makes $2.50 per pound profit on tuna and $2.80 per pound profit for salmon. The market can spend only $106 per day to buy fish and $13 per day to clean it. How much of each type of fish should the market buy to maximize profit? a. Write an objective function P and constraints for a linear program to model the problem. b. Graph the constraint and find the coordinates of each vertex. c. Evaluate P at each vertex to find the maximum profit.

4 Other 23. Explain how to determine whether a system is independent, dependent, or inconsistent without graphing. 2. Explain how to solve a system of equations by substitution.

5 Algebra II - Chapter 3 Review Answer Section MULTIPLE CHOICE 1. ANS: C REF: 3-1 Graphing Systems of Equations TOP: 3-1 Example 3 2. ANS: A REF: 3-2 Solving Systems Algebraically TOP: 3-2 Example 5 3. ANS: B REF: 3-3 Systems of Inequalities SHORT ANSWER. ANS: y 8 8 O 8 x 8 ( 5, ) REF: 3-1 Graphing Systems of Equations TOP: 3-1 Example 1 5. ANS: y 2 2 O 2 x 2 (2, 0)

6 REF: 3-1 Graphing Systems of Equations TOP: 3-1 Example 1 6. ANS: a. b. 6 REF: 3-1 Graphing Systems of Equations TOP: 3-1 Example 2 7. ANS: inconsistent REF: 3-1 Graphing Systems of Equations TOP: 3-1 Example 3 8. ANS: (2, 2) REF: 3-2 Solving Systems Algebraically TOP: 3-2 Example 1 9. ANS: ; 13 adults, 39 children REF: 3-2 Solving Systems Algebraically TOP: 3-2 Example ANS: (0, 2) REF: 3-2 Solving Systems Algebraically TOP: 3-2 Example 11. ANS: no solutions REF: 3-2 Solving Systems Algebraically TOP: 3-2 Example ANS: infinite solutions REF: 3-2 Solving Systems Algebraically TOP: 3-2 Example ANS: y O 2 6 x 2 6

7 REF: 3-3 Systems of Inequalities TOP: 3-3 Example 2 1. ANS: y 2 2 O 2 x 2 REF: 3-3 Systems of Inequalities TOP: 3-3 Example ANS: y 2 2 O 2 x 2 REF: 3-3 Systems of Inequalities TOP: 3-3 Example ANS: Let x = the number of vanilla cakes. Let y = the number of chocolate cakes.

8 y x REF: 3-3 Systems of Inequalities TOP: 3-3 Example ANS: 100 y x REF: 3-3 Systems of Inequalities TOP: 3-3 Example ANS: maximum value at (9, 0); 27 REF: 3- Linear Programming TOP: 3- Example ANS: (0, 2), (2, 0), (, 6); maximum value of 8 REF: 3- Linear Programming TOP: 3- Example ANS: 0 of type A 60 of type B

9 REF: 3- Linear Programming TOP: 3- Example ANS: 10 y Vertices (0, 0): (0, 2): (3, 0): (3, 5): x When x = 3 and y = 5, P has its maximum value of 270. REF: 3- Linear Programming TOP: 3- Example 1 ESSAY 22. ANS: [] a. Let x be pounds of tuna and y be pounds of salmon. The objective function is and the constraints are

10 b. c. The market should buy 28 pounds of tuna and 6 pounds of salmon to maximize profit. [3] two parts correct [2] one part correct [1] correct answers, but no work shown REF: 3- Linear Programming TOP: 3- Example 2 OTHER 23. ANS: Answers may vary. Sample: If the slopes of the two equations are not the same, then the system is independent. If the slopes are the same and the y-intercepts are also the same, then the system is dependent. If the slopes are the same but the y-intercepts are different, then the system is inconsistent. REF: 3-1 Graphing Systems of Equations TOP: 3-1 Example 3 2. ANS: Answers may vary. Sample: Solve one equation for x or y. Substitute the expression for x (or y) into the other equation. You will now have an equation in one variable. Solve for that variable. Use the value of that variable to substitute into either of the first two equations to find the value of the second variable. REF: 3-2 Solving Systems Algebraically TOP: 3-2 Example 1

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