1. Table of Contents Solve Systems by Graphing Solve Systems by Substitution Solve Systems by Elimination Choosing your Strategy Writing Systems to Model Situations Solving Systems of Inequalities Teacher note: Many of the Responder Questions have boxes over the answers. Have the students take some time to solve the system prior to showing them the multiple choice answers by clicking on the box, so they do not just substitute in the answers. Teacher note Click on topic to go to that section.

2. Strategy One: Graphing Return to Table of Contents

3. Some vocabulary... The "solution" to a system is an ordered pair that will work in each equation. One way to find the solution is to graph the equations on the same coordinate plane and find the point of intersection. A "system" is two or more linear equations.

4. Suppose you are walking to school. Your friend is 5 blocks ahead of you. You can walk two blocks per minute, your friend can walk one block per minute. How many minutes will it take for you to catch up with your friend? Consider this...

5. Time (min. ) Friend's distance from your start (blocks) Your distance from your start (blocks) 0 5 0 1 6 2 2 7 4 3 8 6 4 9 8 5 10 First, make a table to represent the problem.

6. Next, plot the points on a graph. Time (min.) Blocks 0 5 20 15 10 1510 5 0 Tim e (m in.) Friend's distanc e from your start (blocks) Your distance from your start(bloc ks) 0 5 0 1 6 2 2 7 4 3 8 6 4 9 8 5 10

7. The point where they intersect is the solution to the system. Time (min.) Blocks 0 5 20 15 10 1510 5 0 (5,10) is the solution. In the context of the problem this means after 5 minutes, you will meet your friend at block 10.

8. Solve the system of equations graphically. y = 2x -3 y = x - 1 Solution Solution (2, 1)

9. Solve the system of equations graphically. 2x + y = 3 x - 2y = 4 Solution Solution (2, -1)

10. Solve the system of equations graphically. 3x + y = 11 x - 2y = 6 Solution Solution (4, -1)

11. Solve using graphing y = 4x+6 move y = -3x-1 move Write the equation for the green dashed line Write the equation for the blue solid line What is this point of intersection? (move the hand!) (-1, 2)

12. (, ) 2 y = 4x+6 y = -3x-1 Now take the ordered pair we just found and substitute it into the equation to prove that it is a solution for both lines.

13. y = 2x + 3 Solve by Graphing y = -4x - 3 (-1,1)

14. y= x - 4y= -3x + 4 Solve by Graphing (2,-2)

15. What's the problem here? y= 2x - 4y= 2x + 4 Parallel lines do not intersect! Therefore there is no solution. No ordered pair that will work in BOTH equations ( ) click to reveal

16. 2y = -4x + 10 2 y = -2x + 5 2x + y = 5 -2x -2x y = -2x + 5 Solve by Graphing Fi rst - transform the equations into y = mx + b form (slope-intercept form) Now graph the two transformed lines.

17. 2y = 10 -4x becomes y = -2x + 5 2x + y = 5 becomes y = -2x + 5 What's the problem? The equations transform to the same line. So we have infinitely many solutions. click to reveal

18. 1 Solve the system by graphing. y = -x + 4 y = 2x +1 A(3,1) B (1,3) C (-1,3) Dno solution Click for multiple choice answers. Solution Solution (1,3)

19. 2 Solve the system by graphing. y = 0.5x - 1 y = -0.5x -1 A (0,-1) B(0,0) C infinitely many Dno solution Solution Solution (0, -1) Click for multiple choice answers.

20. 3 Solve the system by graphing. 2x + y = 3 x - 2y = 4 A (2,4) B(0.4, 2.2) C(2, -1) Dno solution Solution Solution (2, -1) Click for multiple choice answers.

21. 4 Solve the system by graphing. y = 3x + 3 y = 3x - 3 A (0,0) B (3,3) C infinitely many Dno solution Solution No Solution Click for multiple choice answers.

22. 5 Solve the system by graphing. y = 3x + 4 4y = 12x + 16 A(3,4) B (-3,-4) C infinitely many Dno solution Solution Solution - ∞ many Click for multiple choice answers.

23. 6 On the accompanying set of axes, graph and label the following lines: y=5 x = - 4 y = x+5 Calculate the area, in square units, of the triangle formed by the three points of intersection. From the New York State Education Department. Office of Assessment Policy, Development and Administration. Internet. Available from www.nysedregents.org/IntegratedAlgebra; accessed 17, June, 2011. Solution Solution - 10 sq units

24. Strategy Two: Substitution Return to Table of Contents

25. Solve the system of equations graphically. y = x + 6.1 y = -2x - 1.4 NOTE This is not easily solved by graphing. Allow students a few minutes to attempt then stop them and begin the discussion.

26. Graphing can be inefficient or approximate. Another way to solve a system is to use substitution. Substitution allows you to create a one variable equation. Substitution Explanation

27. Solve the system using substitution. Why was it difficult to solve this system by graphing? y = x + 6.1 y = -2x - 1.4 y = -2x - 1.4-start with one equation x + 6.1 = -2x - 1.4-substitute x + 6.1 for y in equation +2x -6.1 +2x - 6.1 3x = -7.5-solve for x x = -2.5 Substitute -2.5 for x in either equation and solve for y. y = x + 6.1 y = (-2.5) + 6.1 y = 3.6 Since x = -2.5 and y = 3.6, the solution is (-2.5, 3.6) CHECK: See if (-2.5, 3.6) satisfies the other equation. y = -2x - 1.4 3.6 = -2(-2.5) - 1.4 3.6 = 5 - 1.4 3.6 = 3.6 ? ?

28. + 3x = 21 -3 y y = -2x +14 Solve the system using substitution. () Solution -3 (-2x + 14) + 3x = 21y = -2(7) + 14 6x - 42 + 3x = 21y = -14 + 14 9x - 42 = 21y = 0 9x = 63 x = 7(7, 0) NOTE Equations can be moved on the page to show substitution into the y of the second equation.

29. = -y - 3 x x = -5y - 39 Solve the system using substitution. () Solution -y - 3 = -5y - 39x = -y -3 4y - 3 = -39x = -(-9) - 3 4y = -36x = 9 -3 y = -9x = 6 (6, -9)

30. Examine each system of equations. Which variable would you choose to substitute? Why? y = 4x - 9.6 y = -2x + 9 y = -3x 7x - y = 42 y = 4x + 1 x = 4y + 1

31. 7Examine the system of equations. Which variable would you substitute? 2x + y = 5 2y = 10 - 4x Ax By Solution y

32. 8Examine the system of equations. Which variable would you substitute? 2y - 8 = x y + 2x = 4 Ax By Solution x

33. 9Examine the system of equations. Which variable would you substitute? x - y = 20 2x + 3y = 0 Ax By Solution x

34. Sometimes you need to rewrite one of the equations so that you can use the substitution method. For example: The system:Is equivalent to: 3x -y = 5y = 3x -5 2x + 5y = -82x + 5y = -8 Using substitution you now have: 2x + 5(3x-5) = -8 -solve for x 2x + 15x - 25 = -8-distribute the 5 17x - 25 = -8-combine x's 17x = 17-at 25 to both sides x = 1- divide by 17 Substitute x = 1 into one of the equations. 2(1) + 5y = -8 2 + 5y = -8 5y = -10 y = -2 The ordered pair (1,-2) satisfies both equations in the original system. 3x -y = 5 2x + 5y = -8 3(1) - (-2) = 52(1) + 5(-2) = -8 3 + 2 = 5 2 - 10 = -8 -8 = -8

35. Your class of 22 is going on a trip. There are four drivers and two types of vehicles, vans and cars. The vans seat six people, and the cars seat four people, including drivers. How many vans and cars does the class need for the trip? Let v = the number of vans and c = the number of cars

36. Set up the system: Drivers: v + c = 4 People: 6v + 4c = 22 Solve the system by substitution. v + c = 4 -solve the first equation for v. v = -c + 4 -substitute -c + 4 for v in the 6(-c + 4) + 4c = 22 second equation -6c + 24 + 4c = 22 -solve for c -2c + 24 = 22 -2c = -2 c = 1 v + c = 4 v + 1 = 4-substitute for c in the 1st equation v = 3-solve for v Since c = 1 and v = 3, they should use 1 car and 3 vans. Check the solution in the equations: v + c = 46v + 4c = 22 3 + 1 = 4 6(3) + 4(1) = 22 4 = 4 18 + 4 = 22 22 = 22

37. Now solve this system using substitution. What happens? x + y = 6 5x + 5y = 10 x + y = 6-solve the first equation for x x = 6 - y 5(6 - y) + 5y = 10-substitute 6 - y for x in 2nd equation 30 - 5y + 5y = 10-solve for y 30 = 10-FALSE! Since 30 = 10 is a false statement, the system has no solution.

38. Now solve this system using substitution. What happens? x + 4y = -3 2x + 8y = -6 x + 4y = -3- solve the first equation for x x = -3 - 4y 2(-3 - 4y) + 8y = -6- sub. -3 - 4y for x in 2nd equation -6 - 8y + 8y = -6- solve for y -6 = -6 - TRUE! - there are infinitely many solutions

39. How can you quickly decide the number of solutions a system has? 1 Solution Different slopes No Solution Same slope; different y- intercept (Parallel Lines) Infinitely Many Same slope; same y-intercept (Same Line)

40. 10 3x - y = -2 y = 3x + 2 A 1 solution Bno solution Cinfinitely many solutions Solution C

41. 113x + 3y = 8 y = x A 1 solution Bno solution Cinfinitely many solutions 1 3 Solution A

42. 12y = 4x 2x - 0.5y = 0 A 1 solution Bno solution Cinfinitely many solutions Solution C

43. 13 3x + y = 5 6x + 2y = 1 A 1 solution Bno solution Cinfinitely many solutions Solution B

44. 14y = 2x - 7 y = 3x + 8 A 1 solution Bno solution Cinfinitely many solutions Solution A

45. 15Solve each system by substitution. y = x - 3 y = -x + 5 A (4,9) B(-4,-9) C(4,1) D (1,4) Solution C Click for multiple choice answers.

46. 16Solve each system by substitution. y = x - 6 y = -4 A(-10,-4) B(-4,2) C(2,-4) D (10,4) Click for multiple choice answers. Solution C

47. 17Solve each system by substitution. y + 2x = -14 y = 2x + 18 A(1,20) B (1,18) C(8,-2) D(-8,2) Click for multiple choice answers. Solution D

48. 18Solve each system by substitution. 4x = -5y + 50 x = 2y - 7 A(6,6.5) B (5,6) C (4,5) D (6,5) Solution B Click for multiple choice answers.

49. 19Solve each system by substitution. y = -3x + 23 -y + 4x = 19 A (6,5) B(-7,5) C (42,-103) D(6,-5) Solution A Click for multiple choice answers.

50. Strategy Three: Elimination Return to Table of Contents

51. When both linear equations of a system are in Standard Form, Ax + By = C, you can solve the system using elimination. You can add or subtract the equations to eliminate a variable.

52. How do you decide which variable to eliminate? First, look to see if one variable has the same or opposite coefficients. If so, eliminate that variable. Second, look for which coefficients have a simple least common multiple. Eliminate that variable.

53. If the variables have the same coefficient, you can subtract the two equations to eliminate the variable. If the variables have opposite coefficients, you add the two equations to eliminate the variable. Sometimes, you need to multiply one, or both, equations by a number in order to create a common coefficient.

54. 5x + y = 44 -4x - y = -34 Solve by Elimination - Click on the terms to eliminate and they will disappear, then add the two equations together. ) (

55. 3x + y = 15 -3x -3y = -21 Solve by Elimination - Click on the terms and they will disappear then add the two equations together. ( )

56. 5x + y = 17 -2x + y = -4 Solve by Elimination - There are 2 ways to complete this problem. See both examples. () 5x + y = 17 2x - y = 4 ( ) - 7x = 21 Multiplication by -1 One method is to recognize that the y-coefficient is the same. You can multiply the second equation by -1. This will create opposite coefficients for the y variable. Then, add the two equations. Subtraction One method is to recognize that the y-coefficient is the same. You can subtract the second equation from the first. However, you must remember to change the sign for every term! 5x + y = 17 -2x + y = -4

57. Solve the system by elimination. 4x + 3y = 16 2x - 3y = 8 Pull 4x + 3y = 164(4) + 3y = 16 + 2x - 3y = 8 16 + 3y = 16 6x = 24 3y = 0 x = 4 y = 0 (4, 0)

58. 20 A (5,1) B (-5,-1) C(1,5) Dno solution Solve each system by elimination. x + y = 6 x - y = 4 Solution A Click for multiple choice answers.

59. 21 Solve each system by elimination. 2x + y = -5 2x - y = -3 A(-2,1) B (-1,-2) C (-2,-1) Dinfinitely many Solution C Click for multiple choice answers.

60. 22 Solve each system by elimination. 2x + y = -6 3x + y = -10 A (4,2) B(3,5) C (2,4) D(-4,2) Solution D Click for multiple choice answers.

61. 23 Solve each system by elimination. 4x - y = 5 x - y = -7 A no solution B (4,11) C(-4,-11) D (11,-4) Solution B Click for multiple choice answers.

62. 24 Solve each system by elimination. 3x + 6y = 48 -5x + 6y = 32 A(2,-7) B (7,2) C(2,7) Dinfinitely many Solution C Click for multiple choice answers.

63. Sometimes, it is not possible to eliminate a variable by adding or subtracting the equations. When this is the case, you need to multiply one or both equations by a nonzero number in order to create a common coefficient. Then add or subtract the equations.

64. Examine each system of equations. Which variable would you choose to eliminate? What do you need to multiply each equation by? 2x + 5y = -1 x + 2y = 0 3x + 8y = 81 5x - 6y = -39 3x + 6y = 6 2x - 3y = 4

65. In order to eliminate the y, you need to multiply first. 3x + 4y = -10 5x - 2y = 18 Multiply the second equation by 2 so the coefficients are opposites. 2(5x - 2y = 18) 10x - 4y = 36 Now solve by adding the equations together. 3x + 4y = -10 10x - 4y = 36 13x = 26 x = 2 Solve for y, by substituting x = 2 into one of the equations. 3x + 4y = -10 3(2) + 4y = -10 6 + 4y = -10 4y = -16 y = -4 So (2,-4) is the solution. Check: 3x + 4y = -10 5x - 2y = 18 3(2) + 4(-4) = -10 5(2) - 2(-4) = 18 6 + -16 = -10 10 + 8 = 18 -10 = -10 18 = 18 +

66. Now solve the same system by eliminating x. What do you multiply the two equations by? 3x + 4y = -10 5x - 2y = 18 Multiply the first equation by 5 and the second equation by 3 so the coefficients will be the same 5(3x + 4y = -10)3(5x - 2y = 18) 15x + 20y = -50 15x - 6y = 54 Now solve by subtracting the equations. 15x + 20y = -50 15x - 6y = 54 26y = -104 y = -4 Solve for x, by substituting y = -4 into one of the equations. 3x + 4y = -10 3x + 4(-4) = -10 3x + -16 = -10 3x = 6 x = 2 So (2,-4) is the solution. Check: 3x + 4y = -10 5x - 2y = 18 3(2) + 4(-4) = -10 5(2) - 2(-4) = 18 6 + -16 = -10 10 + 8 = 18 -10 = -10 18 = 18 -

67. 25 Which variable can you eliminate with the least amount of work? Ax By 9x + 6y = 15 -4x + y = 3 Solution B

68. 26 Which variable can you eliminate with the least amount of work? Ax By 3x - 7y = -2 -6x + 15y = 9 Solution A

69. 27 Which variable can you eliminate with the least amount of work? Ax By x - 3y = -7 2x + 6y = 34 Solution B

70. 28 What will you multiply the first equation by in order to solve this system using elimination? 2x + 5y = 20 3x - 10y = 37 Now solve it.... (11, ) 2525 - click for answer

71. (-5,-2) 3x + 2y = -19 x - 12y = 19 Now solve it.... 29 What will you multiply the first equation by in order to solve this system using elimination? click for answer

72. x + 3y = 4 3x + 4y = 2 Now solve it.... (-2,2) 30 What will you multiply the first equation by in order to solve this system using elimination? click for answer

73. Choose Your Strategy Return to Table of Contents

74. Altogether 292 tickets were sold for a basketball game. An adult ticket costs $3. A student ticket costs $1. Ticket sales were $470. Let a = adults s = students

75. Set up the system: number of tickets sold: a + s = 292 money collected: 3a + s = 470 First eliminate one variable. a + s = 292 - in both equations s has the same 3a + s = 470 coefficient so you subtract the 2 -2a+ 0 = -178 equations in order to eliminate it. a = 89-solve for a Then, find the value of the eliminated variable. a + s = 292 89 + s = 292-substitute 89 for a in 1st equation s = 203-solve for s There were 89 adult tickets and 203 student tickets sold. (89, 203) Check: a + s = 292 3a + s = 470 89 + 203 = 2923(89) + 203 = 470 292 = 292 267 + 203 = 470 470 = 470 -( )

76. 31 A piece of glass with an initial temperature of 99 F is cooled at a rate of 3.5 F/min. At the same time, a piece of copper with an initial temperature of 0 F is heated at a rate of 2.5 F/min. Let m = the number of minutes and t = the temperature in F. Which system models the given information? A Solution B BC t = 99 + 3.5m t = 0 + 2.5m t = 99 - 3.5m t = 0 + 2.5m t = 99 + 3.5m t = 0 - 2.5m

77. 32Which method would you use to solve the system? Agraphing Bsubstitution Celimination t = 99 - 3.5m t = 0 + 2.5m Now solve it... m = 16.5 t = 41.25 This means that in 16.5 minutes, the temperatures will both be 41.25 ℃. click for answer click for equations

78. 33What method would you choose to solve the system? Agraphing Bsubstitution Celimination 4s - 3t = 8 t = -2s -1 Solution B

79. D(-2, ) 34Now solve the system! A (, -2) 4s - 3t = 8 t = -2s -1 1 2 B(, 2) 1 2 C (2, -2) 1 2 Solution A Click for multiple choice answers.

80. 35 What method would you choose to solve the system? Agraphing Bsubstitution C elimination y = 3x - 1 y = 4x Solution B

81. 36 Now solve it! A(1, 4) B(-4, -1) C (-1, 4) y = 3x - 1 y = 4x D (-1, -4) Solution D Click for multiple choice answers.

82. 37 What method would you choose to solve the system? A graphing B substitution C elimination 3m - 4n = 1 3m - 2n = -1 Solution C

83. 38 Now solve it! A (-2, -1) B (-1, -1) C (-1, 1) 3m - 4n = 1 3m - 2n = -1 D (1, 1) Solution B Click for multiple choice answers.

84. 39 What method would you choose to solve the system? A graphing B substitution C elimination y = -2x y = -0.5x + 3 Solution B

85. 40 Now solve it! A (-6, 12) B (2, -4) y = -2x y = -0.5x + 3 C (-2, 4) D (1, -2) Solution C Click for multiple choice answers.

86. 41 What method would you choose to solve the system? A graphing B substitution C elimination 2x - y = 4 x + 3y = 16 Solution C

87. 42 Now solve it! A (6, 5) B (-4, 7) C (-4, 4) 2x - y = 4 x + 3y = 16 D (4, 4) Click for multiple choice answers. Solution D Click for multiple choice answers.

88. 43 What method would you choose to solve the system? A graphing B substitution C elimination u = 4v 3u - 3v = 7 Solution B

89. 44 Now solve it! A(, ) B C (28, 7) u = 4v 3u - 3v = 7 D (7, ) 7 4 28 9 28 9 7 9 7 9 Solution A Click for multiple choice answers.

90. 45Choose a strategy and then answer the question. What is the value of the y-coordinate of the solution to the system of equations x − 2y = 1 and x + 4y = 7? A 1 B C 3 D 4 From the New York State Education Department. Office of Assessment Policy, Development and Administration. Internet. Available from www.nysedregents.org/IntegratedAlgebra; accessed 17, June, 2011. Solution A

91. Modeling Situations Return to Table of Contents

92. A group of 148 people is spending five days at a summer camp. The cook ordered 12 pounds of food for each adult and 9 pounds of food for each child. A total of 1,410 pounds of food was ordered. Part A: Write an equation or a system of equations that describes the above situation and define your variables. From the New York State Education Department. Office of Assessment Policy, Development and Administration. Internet. Available from www.nysedregents.org/IntegratedAlgebra; accessed 17, June, 2011. Pull A = Number of Adults C = Number of Children A + C = 148 12A + 9C = 1,410

93. Part B: Using your work from part A, find: (1) the total number of adults in the group (2) the total number of children in the group Pull 26 Adults, 122 Children

94. Tanisha and Rachel had lunch at the mall. Tanisha ordered three slices of pizza and two colas. Rachel ordered two slices of pizza and three colas. Tanisha’s bill was $6.00, and Rachel’s bill was $5.25. What was the price of one slice of pizza? What was the price of one cola? Pull 3P + 2C = 6.00 2P + 3C = 5.25 Cola: $0.75 Pizza: $1.50 From the New York State Education Department. Office of Assessment Policy, Development and Administration. Internet. Available from www.nysedregents.org/IntegratedAlgebra; accessed 17, June, 2011.

95. Sharu has $2.35 in nickels and dimes. If he has a total of thirty-two coins, how many of each coin does he have? Pull n + d = 32.05n +.10d = 2.35 17 nickels and 15 dimes From the New York State Education Department. Office of Assessment Policy, Development and Administration. Internet. Available from www.nysedregents.org/IntegratedAlgebra; accessed 17, June, 2011.

96. Ben had twice as many nickels as dimes. Altogether, Ben had $4.20. How many nickels and how many dimes did Ben have? From the New York State Education Department. Office of Assessment Policy, Development and Administration. Internet. Available from www.nysedregents.org/IntegratedAlgebra; accessed 17, June, 2011. Pull n = 2d.05n +.10d = 4.20 21 dimes, 42 nickels

97. 46 Your class receives $1105 for selling 205 packages of greeting cards and gift wrap. A pack of cards costs $4 and a pack of gift wrap costs $9 Set up a system and solve. How many packages of cards were sold? You will answer how many packages of gift wrap in the next question. Solution 148 pks of cards

98. 47 Your class receives $1105 for selling 205 packages of greeting cards and gift wrap. A pack of cards costs $4 and a pack of gift wrap costs $9 Set up a system and solve. How many packages of gift wrap were sold? 57 pks of gift wrap Solution

99. 48 The sum of two numbers is 47, and their difference is 15. What is the larger number? A 16 B 31 C 32 D 36 From the New York State Education Department. Office of Assessment Policy, Development and Administration. Internet. Available from www.nysedregents.org/IntegratedAlgebra; accessed 17, June, 2011. B Solution

100. 49 Ramon rented a sprayer and a generator. On his first job, he used each piece of equipment for 6 hours at a total cost of $90. On his second job, he used the sprayer for 4 hours and the generator for 8 hours at a total cost of $100. What was the hourly cost for the sprayer? From the New York State Education Department. Office of Assessment Policy, Development and Administration. Internet. Available from www.nysedregents.org/IntegratedAlgebra; accessed 17, June, 2011. $ 5 per hour Solution

101. 50What is true of the graphs of the two lines 3y - 8 = -5x and 3x = 2y -18? Ano intersection Bintersect at (2,-6) Cintersect at (-2,6) Dare identical C Solution

102. 51 You have 15 coins in your pocket that are either quarters or nickels. They total $2.75. Set up a system to solve. Which method will you use? (Solving it comes later...) Agraphing Bsubstitution Celimination B Solution

103. 52 You have 15 coins in your pocket that are either quarters or nickels. They total $2.75. How many quarters do you have? 10 quarters Solution

104. 53 You have 15 coins in your pocket that are either quarters or nickels. They total $2.75. How many nickels do you have? 5 nickels Solution

105. 54 Julia went to the movies and bought one jumbo popcorn and two chocolate chip cookies for $5.00. Marvin went to the same movie and bought one jumbo popcorn and four chocolate chip cookies for $6.00. How much does one chocolate chip cookie cost? A $0.50 B $0.75 C $1.00 D $2.00 From the New York State Education Department. Office of Assessment Policy, Development and Administration. Internet. Available from www.nysedregents.org/IntegratedAlgebra; accessed 17, June, 2011. A Solution

106. 55 Mary and Amy had a total of 20 yards of material from which to make costumes. Mary used three times more material to make her costume than Amy used, and 2 yards of material was not used. How many yards of material did Amy use for her costume? From the New York State Education Department. Office of Assessment Policy, Development and Administration. Internet. Available from www.nysedregents.org/IntegratedAlgebra; accessed 17, June, 2011. 4.5 yards Solution

107. 56 The tickets for a dance recital cost $5.00 for adults and $2.00 for children. If the total number of tickets sold was 295 and the total amount collected was $1220, how many adult tickets were sold? From the New York State Education Department. Office of Assessment Policy, Development and Administration. Internet. Available from www.nysedregents.org/IntegratedAlgebra; accessed 17, June, 2011. 210 adult tickets Solution

108. Solving Systems of Inequalities Return to Table of Contents

109. Two or more linear inequalities form a system of inequalities. A solution to the system is an ordered pair that is a solution of each inequality in the system. Since inequalities have more than one solution, the solutions are best shown in a graph.

110. Graphing a System of Linear Inequalities 1. Graph the boundary lines of each inequality. (Remember use a dashed line for and a solid line for ) 2. Shade the half-plane for each inequality. 3. The intersection of the half-planes is the solution.

111. Solve the system of inequalities. x + 2y < 6 -x + y < 0 Pull

112. Solve the system of inequalities. 2x + y > -4 x - 2y < 4 Pull

113. Solve the system of inequalities. 4x + 2y < 8 4x + 2y > -8 Pull

114. Ungroup and type text here. Graph the following system of inequalities on the set of axes shown below and label the solution set S. y > −x + 2 y ≤ 2 x + 5 3

115. A company manufactures bicycles and skateboards. The company’s daily production of bicycles cannot exceed 10, and its daily production of skateboards must be less than or equal to 12. The combined number of bicycles and skateboards cannot be more than 16. If x is the number of bicycles and y is the number of skateboards, graph on the accompanying set of axes the region that contains the number of bicycles and skateboards the company can manufacture daily. Pull

116. Write a system of inequalities from the graph. Pull 3x + 4y < 12 y > 2

117. 57Solve the system of linear inequalities. ABC y > -2x + 1 y < x + 2 B Solution

118. 58Solve the system of linear inequalities. ABC x > 2 y < 5 C Solution

119. 59Solve the system of linear inequalities. ABC -2x - 2y < 4 y - 2x > 1 A Solution

120. 60Solve the system of linear inequalities. ABC -5x + y > -2 4x + y < 1 A Solution

121. 61Solve the system of linear inequalities. ABC 3x + 2y < 12 2x - 2y < 20 A Solution

122. 62 A (0,4) B (2,4) C (-4,1) D (4,-1) Which point is in the solution set of the system of inequalities shown in the accompanying graph? From the New York State Education Department. Office of Assessment Policy, Development and Administration. Internet. Available from www.nysedregents.org/IntegratedAlgebra; accessed 17, June, 2011. C Solution

123. 63 Which ordered pair is in the solution set of the system of inequalities shown in the accompanying graph? A (0, 0) B (0, 1) C (1, 5) D (3, 2) From the New York State Education Department. Office of Assessment Policy, Development and Administration. Internet. Available from www.nysedregents.org/IntegratedAlgebra; accessed 17, June, 2011. D Solution

124. 64 Which ordered pair is in the solution set of the following system of linear inequalities? A (0,3) B (2,0) C (−1,0) D (−1,−4) y < 2x + 2 y ≥ −x − 1 From the New York State Education Department. Office of Assessment Policy, Development and Administration. Internet. Available from www.nysedregents.org/IntegratedAlgebra; accessed 17, June, 2011. B Solution

125. 65 Mr. Braun has $75.00 to spend on pizzas and soda for a picnic. Pizzas cost $9.00 each and the drinks cost $0.75 each. Five times as many drinks as pizzas are needed. What is the maximum number of pizzas that Mr. Braun can buy? From the New York State Education Department. Office of Assessment Policy, Development and Administration. Internet. Available from www.nysedregents.org/IntegratedAlgebra; accessed 17, June, 2011. 5 pizzas Solution