MATH 021 TEST 3 REVIEW SHEET

Transcription

1 TO THE STUDENT: MATH 021 TEST 3 REVIEW SHEET This Review Sheet gives an outline of the topics covered on Test 3 as well as practice problems. Answers for all problems begin on page 8. In several instances, it is suggested that you go over some of the definitions used in this course. Definitions can be found in the Glossary/Index listed on pages A-49 to A-56 of your textbook. I. Quantity-Rate Problems For problems #1 5, set up a quantity-rate table. Then use the table to solve the problem. 1. A record store sells new CD s for $12 each and used CD s for $8 each. If I want to buy 7 more used CD s than new ones, how many of each kind can I buy with $236? 2. Joella told her brother that she earned $650 in interest that year. She started the year with $10,000 and invested part of it in stocks at a 7.5% interest rate and she invested the rest in bonds at a 5% interest rate. Determine how much she invested in stocks and how much in bonds. 3. How many gallons of hot water at 135 F would be need to be added to a bathtub containing 18 gallons of cold water at 50 F to raise the temperature to 104 F? 4. Brian has a blend of coffee beans that combines Mocha Java, worth $8.50 per pound, with Costa Rican, worth $9.25 per pound. He wishes to blend 200 pounds. How many pounds of each does he need to obtain 200 pounds to sell at $9.10 per pound? 5. Over the last two semesters, Lisa completed 24 credits with a GPA of 3.2. This semester she earned an A in her 3-credit history class, a C in her 4-credit science class, a C in her 3-credit psychology class, and an A in her 2-credit physical education class. What is Lisa s new GPA? II. Solving Linear Systems 6. Fill in the Blank: A solution to a system of two equations in two variables is an ordered that makes equations. 1

2 7. True or False: Every system of two linear equations has exactly one solution. Explain your answer. 8. Solve each system of equations by graphing. Use these steps: Graph both equations on the same coordinate system. Label each line with its equation. Write the solution. Check the solution by substituting it back into both equations. a. 9. Use the graph to answer the following questions. a. Write the equation of Line 1 and the equation of Line 2. Write the system of equations represented by the graph. Identify the solution of the system from the graph. d. Check the solution in each equation. 10. Solve each system of equations by substitution. a. d. 2

3 11. Solve each system of equations by elimination. a. d. III. Applications For problems #12 15, solve each problem using an algebraic approach. You may need a quantity-rate table. You may need to draw a diagram. You may wish to set up a system of two equations in two variables or one equation in one variable. Use whatever tool is appropriate for the problem. Make sure to define each variable you use. 12. A Boeing 747 holds 279 more people than a Boeing 707. If the two planes together carry 721 people (including crew), how many does each plane carry? 13. An English muffin and two fried eggs contain 330 calories. Three English muffins and one fried egg contain 515 calories. How many calories are in one English muffin and one fried egg? 14. Janet has borrowed a total of $3500. Some of it is an 8% auto loan and the rest is at 21% on her credit card. She pays $345 per year interest on the loans. How much did she borrow from each source? 15. A vegetable garden is in the shape of a rectangle in which the length is 3 feet more than twice the width. If the perimeter of the garden is 72 feet, what is the length and width of the garden? IV. Simplifying Expressions 16. Consider the expression: a. Identify the constant term. What is the degree of the polynomial? Because this expression has terms, it is called a. 3

4 17. Simplify each expression. a. d. e. f. 18. Simplify completely. Show work using the properties of radicals. a. 19. Multiply and simplify as needed. a. d. e. f. 20. Simplify completely. a. d. V. Review from Units 1 and 2 (Note: In addition to the following problems, you should also review your notes and other materials from Units 1 and 2.) 21. Find the slope of the line through each pair of points. Then determine whether the line is rising falling, vertical, or horizontal. a. 4

5 22. On graph paper, graph the line with slope 3 that passes through (2, 1). 23. Suppose: a. Write the equation in slope-intercept form. Identify the slope and y-intercept. Use the slope and y-intercept to graph the line (on graph paper). 24. Suppose: a. Find the x-intercept. Write your answer as an ordered pair. Find the y-intercept. Write your answer as an ordered pair. Use the intercept points to graph the line (on graph paper). 25. Write the equation of the line with slope 5 and y-intercept. 26. Write the equation of the line with slope which passes through the point. 27. Write the equation of the line containing the points. 28. A line goes through the points. Write the equation of this line. 29. A vertical line goes through the point (12, 18). Write the equation of this line. 30. Determine whether each pair of lines is parallel. Explain how you know. a. and and 31. For the following graph, find the slope, y-intercept, and equation of the line. 5

6 32. State whether each graph represents a function? Then explain why. a. 33. Suppose a. Find. Write your answer as an ordered pair. What is the name of this special point? Find x if. Write your answer as an ordered pair. What is the name of this special point? 34. You are buying favors for a child s birthday party. The input is the number of party favors, p. The output is the total cost, C. The cost is a function of the number of party favors purchased, so we write. a. Interpret in the context of this situation. If the cost of 10 party favors is $50.00, write this fact using function notation. 35. Each month, the cost to rent a postage meter is $5 plus $.40 for each letter stamped by the meter. a. In this situation is the total cost the input or the output? The input is the. Total cost is a function of the number of letters stamped because for any there is exactly one. d. What is the slope? What is the practical meaning of the slope in the context of this situation? e. What is the vertical intercept? What is the practical meaning of the vertical intercept in the context of the situation? f. Write the equation of the total cost function. Use C for total cost and n for the number of letters stamped. g. Make an input-output table. Show the total cost for 0 to 30 letters, in steps of 5. h. On graph paper, graph this relationship. Be sure to include an appropriate scale and to label your axes. 6

7 36. Solve each equation, and check your solution. a. d. e. f. 37. Solve each formula for variable indicated. a. 38. Solve each inequality algebraically. Then graph the solution on a number line. a. 39. Evaluate. 40. Solve each equation using the graph of shown below. Name all solutions. a. Approximate the solutions to. 7

8 41. A portion of the table for is given below. Use this table to solve each equation. x a. Estimate the solution to. 42. Write each sentence as an equation with x as the input and y as the output. a. The output is four more than the square root of the input. The output is three less than the square of the input. 43. a. Make an input-output table for the equation. Use integers from 3 to 3 for the input. Based on your table, graph the equation. Be sure to label the axes with an appropriate scale. 8

9 ANSWER KEY Item Q R Q*R New CDs x 12 12x Used CDs x+7 8 8(x+7) Total 236 Item Q R Q*R Mocha Java x x Costa Rican 200-x (200-x) Total I can buy 9 new CDs and 16 used CDs. Brian will need 40 lbs of Mocha Java coffee and 160 lbs of Costa Rican coffee. Item Q R Q*R Stocks x x Bonds x (10000-x) Total Joella invested $6000 in stocks and $4000 in bonds. 5. Item Q R Q*R Last two semesters History Science Psych Phys. Ed Total Item Q R Q*R Hot Water x x Cold Water Total 18+x (18+x) 6. pair, both, true 7. False. A system of two linear equations would have no solution, if the lines were parallel. Or the system could have an infinite number of solutions, if the lines were coincident. You would add approximately 31.4 gallons of hot water. 9

10 8. a. 11. a. No solution d. 12. Let x = the number of passengers in a 747 Let y = the number of passengers in a 707 Solution is: The 747 holds 500 passengers, and the 707 holds 221 passengers. 13. Let m = the number of calories in an English muffin Let e = the number of calories in a fried egg There are 140 calories in an English muffin and 95 calories in a fried egg. No Solution. 9. a. Line 1: Line 2: 14. Let a = the amount of the auto loan Let c = the amount on the credit card d. Line 1: Line 2: Janet borrowed $3000 on the auto loan and $500 on the credit card. 15. Let l = the length of the garden Let w = the width of the garden 10. a. Infinite number of solutions d. The length of the garden is 25 ft, and the width is 11 ft. 16. a , binomial 10

11 17. a. d. e. f. 18. a a. d. e. f. 24. a. 20. a. d. 21. a. ; falling 22. ; horizontal no slope; vertical a. 30. a. These lines are parallel because they have the same slope, which is 3. 11

12 30. These lines are not parallel because they do not have the same slope. The slope of the first line is -6, and the slope of the second 35. h. line is. 31. ; ; 32. a. Not a function. This graph fails the Vertical Line Test. For some input, there is more than one output. This is a function. It passes the Vertical Line Test. For each input, there is exactly one output. 33. a. ; ; y-intercept ; ; x-intercept 34. a. If you purchase 8 party favors, the total cost is $ a. Total cost is the output. number of letters stamped number of letters, total cost d. ; There is a charge of $0.40 per letter. e. ; There is an initial charge of $5.00. f. 35. g. Input: n Number of letters stamped Output: C(n) Total Cost a. d. e. f. 37. a. 38. a. 39. or 40. a. No solution 41. a. x is between -3 and

13 42. a. 43. a. x