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Geometry Section 0-8 11-12

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Geometry Section 0-8 11-12

  1. 1. Section 0-8 Systems of EquationsMonday, September 19, 2011
  2. 2. Essential Question How do you use graphing, substitution, and elimination to solve systems of linear equations?Monday, September 19, 2011
  3. 3. Vocabulary 1. System of Equations: 2. Substitution: 3. Elimination:Monday, September 19, 2011
  4. 4. Vocabulary 1. System of Equations: Two or more equations with the same two variables that you solve at the same time 2. Substitution: 3. Elimination:Monday, September 19, 2011
  5. 5. Vocabulary 1. System of Equations: Two or more equations with the same two variables that you solve at the same time 2. Substitution: Plugging in a number or expression for a variable 3. Elimination:Monday, September 19, 2011
  6. 6. Vocabulary 1. System of Equations: Two or more equations with the same two variables that you solve at the same time 2. Substitution: Plugging in a number or expression for a variable 3. Elimination: Using addition and multiplication to eliminate parts of a system to achieve a solutionMonday, September 19, 2011
  7. 7. Types of SolutionsMonday, September 19, 2011
  8. 8. Types of Solutions One solution: a pointMonday, September 19, 2011
  9. 9. Types of Solutions One solution: a point No solutions: parallel linesMonday, September 19, 2011
  10. 10. Types of Solutions One solution: a point No solutions: parallel lines Infinitely many solutions on the line: same lineMonday, September 19, 2011
  11. 11. Example 1 Solve the system by graphing. y xMonday, September 19, 2011
  12. 12. Example 1 Solve the system by graphing. y ⎧ y = 2x − 4 ⎪ ⎨ x ⎪ y = −x +5 ⎩Monday, September 19, 2011
  13. 13. Example 1 Solve the system by graphing. y ⎧ y = 2x − 4 ⎪ ⎨ x ⎪ y = −x +5 ⎩Monday, September 19, 2011
  14. 14. Example 1 Solve the system by graphing. y ⎧ y = 2x − 4 ⎪ ⎨ x ⎪ y = −x +5 ⎩Monday, September 19, 2011
  15. 15. Example 1 Solve the system by graphing. y ⎧ y = 2x − 4 ⎪ ⎨ x ⎪ y = −x +5 ⎩Monday, September 19, 2011
  16. 16. Example 1 Solve the system by graphing. y ⎧ y = 2x − 4 ⎪ ⎨ x ⎪ y = −x +5 ⎩Monday, September 19, 2011
  17. 17. Example 1 Solve the system by graphing. y ⎧ y = 2x − 4 ⎪ ⎨ x ⎪ y = −x +5 ⎩Monday, September 19, 2011
  18. 18. Example 1 Solve the system by graphing. y ⎧ y = 2x − 4 ⎪ ⎨ x ⎪ y = −x +5 ⎩Monday, September 19, 2011
  19. 19. Example 1 Solve the system by graphing. y ⎧ y = 2x − 4 ⎪ ⎨ x ⎪ y = −x +5 ⎩Monday, September 19, 2011
  20. 20. Example 1 Solve the system by graphing. y ⎧ y = 2x − 4 ⎪ ⎨ x ⎪ y = −x +5 ⎩Monday, September 19, 2011
  21. 21. Example 1 Solve the system by graphing. y ⎧ y = 2x − 4 ⎪ ⎨ x ⎪ y = −x +5 ⎩Monday, September 19, 2011
  22. 22. Example 1 Solve the system by graphing. y ⎧ y = 2x − 4 ⎪ ⎨ x ⎪ y = −x +5 ⎩Monday, September 19, 2011
  23. 23. Example 1 Solve the system by graphing. y ⎧ y = 2x − 4 ⎪ ⎨ x ⎪ y = −x +5 ⎩Monday, September 19, 2011
  24. 24. Example 1 Solve the system by graphing. y ⎧ y = 2x − 4 ⎪ ⎨ x ⎪ y = −x +5 ⎩Monday, September 19, 2011
  25. 25. Example 1 Solve the system by graphing. y ⎧ y = 2x − 4 ⎪ ⎨ x ⎪ y = −x +5 ⎩Monday, September 19, 2011
  26. 26. Example 1 Solve the system by graphing. y ⎧ y = 2x − 4 ⎪ ⎨ x ⎪ y = −x +5 ⎩Monday, September 19, 2011
  27. 27. Example 1 Solve the system by graphing. y ⎧ y = 2x − 4 ⎪ ⎨ x ⎪ y = −x +5 ⎩Monday, September 19, 2011
  28. 28. Example 1 Solve the system by graphing. y ⎧ y = 2x − 4 ⎪ ⎨ x ⎪ y = −x +5 ⎩Monday, September 19, 2011
  29. 29. Example 1 Solve the system by graphing. y ⎧ y = 2x − 4 ⎪ ⎨ x ⎪ y = −x +5 ⎩Monday, September 19, 2011
  30. 30. Example 1 Solve the system by graphing. y ⎧ y = 2x − 4 ⎪ ⎨ x ⎪ y = −x +5 ⎩Monday, September 19, 2011
  31. 31. Example 1 Solve the system by graphing. y ⎧ y = 2x − 4 ⎪ ⎨ x ⎪ y = −x +5 ⎩Monday, September 19, 2011
  32. 32. Example 1 Solve the system by graphing. y ⎧ y = 2x − 4 ⎪ ⎨ x ⎪ y = −x +5 ⎩Monday, September 19, 2011
  33. 33. Example 1 Solve the system by graphing. y ⎧ y = 2x − 4 ⎪ ⎨ x ⎪ y = −x +5 ⎩Monday, September 19, 2011
  34. 34. Example 1 Solve the system by graphing. y ⎧ y = 2x − 4 ⎪ ⎨ x ⎪ y = −x +5 ⎩Monday, September 19, 2011
  35. 35. Example 1 Solve the system by graphing. y ⎧ y = 2x − 4 ⎪ ⎨ x ⎪ y = −x +5 ⎩Monday, September 19, 2011
  36. 36. Example 1 Solve the system by graphing. y ⎧ y = 2x − 4 ⎪ ⎨ x ⎪ y = −x +5 ⎩Monday, September 19, 2011
  37. 37. Example 1 Solve the system by graphing. y ⎧ y = 2x − 4 ⎪ ⎨ x ⎪ y = −x +5 ⎩Monday, September 19, 2011
  38. 38. Example 1 Solve the system by graphing. y ⎧ y = 2x − 4 ⎪ ⎨ x ⎪ y = −x +5 ⎩Monday, September 19, 2011
  39. 39. Example 1 Solve the system by graphing. y (3, 2) ⎧ y = 2x − 4 ⎪ ⎨ x ⎪ y = −x +5 ⎩Monday, September 19, 2011
  40. 40. Example 1 Solve the system by graphing. y (3, 2) ⎧ y = 2x − 4 Check: ⎪ ⎨ x ⎪ y = −x +5 ⎩Monday, September 19, 2011
  41. 41. Example 1 Solve the system by graphing. y (3, 2) ⎧ y = 2x − 4 Check: ⎪ ⎨ x ⎪ y = −x +5 ⎩ 2 = 2(3)− 4Monday, September 19, 2011
  42. 42. Example 1 Solve the system by graphing. y (3, 2) ⎧ y = 2x − 4 Check: ⎪ ⎨ x ⎪ y = −x +5 ⎩ 2 = 2(3)− 4 2 = 6− 4Monday, September 19, 2011
  43. 43. Example 1 Solve the system by graphing. y (3, 2) ⎧ y = 2x − 4 Check: ⎪ ⎨ x ⎪ y = −x +5 ⎩ 2 = 2(3)− 4 2 = 6− 4 2 = −3+5Monday, September 19, 2011
  44. 44. Example 1 Solve the system by graphing. y (3, 2) ⎧ y = 2x − 4 Check: ⎪ ⎨ x ⎪ y = −x +5 ⎩ 2 = 2(3)− 4 2 = 6− 4 2 = −3+5Monday, September 19, 2011
  45. 45. Solve a System of Equations by SubstitutionMonday, September 19, 2011
  46. 46. Solve a System of Equations by Substitution 1. Solve one equation for one variable (your choice)Monday, September 19, 2011
  47. 47. Solve a System of Equations by Substitution 1. Solve one equation for one variable (your choice) 2. Substitute the expression from the equation into the other equationMonday, September 19, 2011
  48. 48. Solve a System of Equations by Substitution 1. Solve one equation for one variable (your choice) 2. Substitute the expression from the equation into the other equation 3. Solve for the variable and substitute back into the original equation to find the other variableMonday, September 19, 2011
  49. 49. Solve a System of Equations by Substitution 1. Solve one equation for one variable (your choice) 2. Substitute the expression from the equation into the other equation 3. Solve for the variable and substitute back into the original equation to find the other variable 4. Rewrite your answer as an ordered pair and check it!Monday, September 19, 2011
  50. 50. Example 2 ⎧x + y = 9 ⎪ ⎨ ⎪10x + y =12x ⎩Monday, September 19, 2011
  51. 51. Example 2 ⎧x + y = 9 ⎪ ⎨ ⎪10x + y =12x ⎩ y = −x + 9Monday, September 19, 2011
  52. 52. Example 2 ⎧x + y = 9 ⎪ ⎨ ⎪10x + y =12x ⎩ y = −x + 9 10x +(−x + 9) =12xMonday, September 19, 2011
  53. 53. Example 2 ⎧x + y = 9 ⎪ ⎨ ⎪10x + y =12x ⎩ y = −x + 9 10x +(−x + 9) =12x 9x + 9 =12xMonday, September 19, 2011
  54. 54. Example 2 ⎧x + y = 9 ⎪ ⎨ ⎪10x + y =12x ⎩ y = −x + 9 10x +(−x + 9) =12x 9x + 9 =12x 9 = 3xMonday, September 19, 2011
  55. 55. Example 2 ⎧x + y = 9 ⎪ ⎨ ⎪10x + y =12x ⎩ y = −x + 9 10x +(−x + 9) =12x 9x + 9 =12x 9 = 3x x =3Monday, September 19, 2011
  56. 56. Example 2 ⎧x + y = 9 ⎪ ⎨ ⎪10x + y =12x ⎩ y = −x + 9 3+ y = 9 10x +(−x + 9) =12x 9x + 9 =12x 9 = 3x x =3Monday, September 19, 2011
  57. 57. Example 2 ⎧x + y = 9 ⎪ ⎨ ⎪10x + y =12x ⎩ y = −x + 9 3+ y = 9 10x +(−x + 9) =12x y =6 9x + 9 =12x 9 = 3x x =3Monday, September 19, 2011
  58. 58. Example 2 ⎧x + y = 9 ⎪ ⎨ ⎪10x + y =12x ⎩ y = −x + 9 3+ y = 9 10x +(−x + 9) =12x y =6 9x + 9 =12x Check: 9 = 3x x =3Monday, September 19, 2011
  59. 59. Example 2 ⎧x + y = 9 ⎪ ⎨ ⎪10x + y =12x ⎩ y = −x + 9 3+ y = 9 10x +(−x + 9) =12x y =6 9x + 9 =12x Check: 9 = 3x 3+ 6 = 9 x =3Monday, September 19, 2011
  60. 60. Example 2 ⎧x + y = 9 ⎪ ⎨ ⎪10x + y =12x ⎩ y = −x + 9 3+ y = 9 10x +(−x + 9) =12x y =6 9x + 9 =12x Check: 9 = 3x 3+ 6 = 9 x =3 10(3)+ 6 =12(3)Monday, September 19, 2011
  61. 61. Example 2 ⎧x + y = 9 ⎪ ⎨ ⎪10x + y =12x ⎩ y = −x + 9 3+ y = 9 10x +(−x + 9) =12x y =6 9x + 9 =12x Check: 9 = 3x 3+ 6 = 9 x =3 10(3)+ 6 =12(3) 30 + 6 = 36Monday, September 19, 2011
  62. 62. Example 2 ⎧x + y = 9 ⎪ ⎨ ⎪10x + y =12x ⎩ y = −x + 9 3+ y = 9 10x +(−x + 9) =12x y =6 9x + 9 =12x Check: (3, 6) 9 = 3x 3+ 6 = 9 x =3 10(3)+ 6 =12(3) 30 + 6 = 36Monday, September 19, 2011
  63. 63. Solve by EliminationMonday, September 19, 2011
  64. 64. Solve by Elimination 1. Choose a variable to eliminate (your choice).Monday, September 19, 2011
  65. 65. Solve by Elimination 1. Choose a variable to eliminate (your choice). 2. Make the coefficients of that variable opposite. You might need to multiply to do this. Then combine equations.Monday, September 19, 2011
  66. 66. Solve by Elimination 1. Choose a variable to eliminate (your choice). 2. Make the coefficients of that variable opposite. You might need to multiply to do this. Then combine equations. 3. Solve for the remaining variable.Monday, September 19, 2011
  67. 67. Solve by Elimination 1. Choose a variable to eliminate (your choice). 2. Make the coefficients of that variable opposite. You might need to multiply to do this. Then combine equations. 3. Solve for the remaining variable. 4. Plug back into an original equation to find the other variable.Monday, September 19, 2011
  68. 68. Solve by Elimination 1. Choose a variable to eliminate (your choice). 2. Make the coefficients of that variable opposite. You might need to multiply to do this. Then combine equations. 3. Solve for the remaining variable. 4. Plug back into an original equation to find the other variable. 5. Check and rewrite the answer.Monday, September 19, 2011
  69. 69. Example 3 Solve by combining the equations ⎧7x + 2 y = 5 ⎪ ⎨ ⎪ 2x + 3 y = 16 ⎩Monday, September 19, 2011
  70. 70. Example 3 Solve by combining the equations ⎧7x + 2 y = 5)(3) ⎪( ⎨ ⎪ 2x + 3 y = 16 ⎩Monday, September 19, 2011
  71. 71. Example 3 Solve by combining the equations ⎧7x + 2 y = 5)(3) ⎪( ⎨ ⎪ 2x + 3 y = 16)(−2) ⎩(Monday, September 19, 2011
  72. 72. Example 3 Solve by combining the equations ⎧7x + 2 y = 5)(3) ⎪( ⎨ ⎪ 2x + 3 y = 16)(−2) ⎩( 21x + 6 y = 15Monday, September 19, 2011
  73. 73. Example 3 Solve by combining the equations ⎧7x + 2 y = 5)(3) ⎪( ⎨ ⎪ 2x + 3 y = 16)(−2) ⎩( 21x + 6 y = 15 −4x − 6 y = −32Monday, September 19, 2011
  74. 74. Example 3 Solve by combining the equations ⎧7x + 2 y = 5)(3) ⎪( ⎨ ⎪ 2x + 3 y = 16)(−2) ⎩( 21x + 6 y = 15 −4x − 6 y = −32Monday, September 19, 2011
  75. 75. Example 3 Solve by combining the equations ⎧7x + 2 y = 5)(3) ⎪( ⎨ ⎪ 2x + 3 y = 16)(−2) ⎩( 21x + 6 y = 15 −4x − 6 y = −32 17x = −17Monday, September 19, 2011
  76. 76. Example 3 Solve by combining the equations ⎧7x + 2 y = 5)(3) ⎪( ⎨ ⎪ 2x + 3 y = 16)(−2) ⎩( 21x + 6 y = 15 −4x − 6 y = −32 17x = −17 17 17Monday, September 19, 2011
  77. 77. Example 3 Solve by combining the equations ⎧7x + 2 y = 5)(3) ⎪( ⎨ ⎪ 2x + 3 y = 16)(−2) ⎩( 21x + 6 y = 15 −4x − 6 y = −32 17x = −17 17 17 x = −1Monday, September 19, 2011
  78. 78. Example 3 Solve by combining the equations ⎧7x + 2 y = 5)(3) ⎪( ⎨ ⎪ 2x + 3 y = 16)(−2) ⎩( 21x + 6 y = 15 2(−1) + 3 y = 16 −4x − 6 y = −32 17x = −17 17 17 x = −1Monday, September 19, 2011
  79. 79. Example 3 Solve by combining the equations ⎧7x + 2 y = 5)(3) ⎪( ⎨ ⎪ 2x + 3 y = 16)(−2) ⎩( 21x + 6 y = 15 2(−1) + 3 y = 16 −4x − 6 y = −32 −2 + 3 y = 16 17x = −17 17 17 x = −1Monday, September 19, 2011
  80. 80. Example 3 Solve by combining the equations ⎧7x + 2 y = 5)(3) ⎪( ⎨ ⎪ 2x + 3 y = 16)(−2) ⎩( 21x + 6 y = 15 2(−1) + 3 y = 16 −4x − 6 y = −32 −2 + 3 y = 16 17x = −17 +2 +2 17 17 x = −1Monday, September 19, 2011
  81. 81. Example 3 Solve by combining the equations ⎧7x + 2 y = 5)(3) ⎪( ⎨ ⎪ 2x + 3 y = 16)(−2) ⎩( 21x + 6 y = 15 2(−1) + 3 y = 16 −4x − 6 y = −32 −2 + 3 y = 16 17x = −17 +2 +2 17 17 3 y = 18 x = −1Monday, September 19, 2011
  82. 82. Example 3 Solve by combining the equations ⎧7x + 2 y = 5)(3) ⎪( ⎨ ⎪ 2x + 3 y = 16)(−2) ⎩( 21x + 6 y = 15 2(−1) + 3 y = 16 −4x − 6 y = −32 −2 + 3 y = 16 17x = −17 +2 +2 17 17 3 y = 18 x = −1 3 3Monday, September 19, 2011
  83. 83. Example 3 Solve by combining the equations ⎧7x + 2 y = 5)(3) ⎪( ⎨ ⎪ 2x + 3 y = 16)(−2) ⎩( 21x + 6 y = 15 2(−1) + 3 y = 16 −4x − 6 y = −32 −2 + 3 y = 16 17x = −17 +2 +2 17 17 3 y = 18 x = −1 3 3 y=6Monday, September 19, 2011
  84. 84. Example 3 Solve by combining the equations ⎧7x + 2 y = 5)(3) ⎪( ⎨ ⎪ 2x + 3 y = 16)(−2) ⎩( Check: 21x + 6 y = 15 2(−1) + 3 y = 16 −4x − 6 y = −32 −2 + 3 y = 16 17x = −17 +2 +2 17 17 3 y = 18 x = −1 3 3 y=6Monday, September 19, 2011
  85. 85. Example 3 Solve by combining the equations ⎧7x + 2 y = 5)(3) ⎪( ⎨ ⎪ 2x + 3 y = 16)(−2) ⎩( Check: 21x + 6 y = 15 2(−1) + 3 y = 16 −4x − 6 y = −32 7(−1) + 2(6) = 5 −2 + 3 y = 16 17x = −17 +2 +2 17 17 3 y = 18 x = −1 3 3 y=6Monday, September 19, 2011
  86. 86. Example 3 Solve by combining the equations ⎧7x + 2 y = 5)(3) ⎪( ⎨ ⎪ 2x + 3 y = 16)(−2) ⎩( Check: 21x + 6 y = 15 2(−1) + 3 y = 16 −4x − 6 y = −32 7(−1) + 2(6) = 5 −2 + 3 y = 16 +2 −7 + 12 = 5 17x = −17 +2 17 17 3 y = 18 x = −1 3 3 y=6Monday, September 19, 2011
  87. 87. Example 3 Solve by combining the equations ⎧7x + 2 y = 5)(3) ⎪( ⎨ ⎪ 2x + 3 y = 16)(−2) ⎩( Check: 21x + 6 y = 15 2(−1) + 3 y = 16 −4x − 6 y = −32 7(−1) + 2(6) = 5 −2 + 3 y = 16 +2 −7 + 12 = 5 17x = −17 +2 17 17 3 y = 18 2(−1) + 3(6) = 16 x = −1 3 3 y=6Monday, September 19, 2011
  88. 88. Example 3 Solve by combining the equations ⎧7x + 2 y = 5)(3) ⎪( ⎨ ⎪ 2x + 3 y = 16)(−2) ⎩( Check: 21x + 6 y = 15 2(−1) + 3 y = 16 −4x − 6 y = −32 7(−1) + 2(6) = 5 −2 + 3 y = 16 +2 −7 + 12 = 5 17x = −17 +2 17 17 3 y = 18 2(−1) + 3(6) = 16 x = −1 3 3 −2 + 18 = 16 y=6Monday, September 19, 2011
  89. 89. Example 3 Solve by combining the equations ⎧7x + 2 y = 5)(3) ⎪( ⎨ ⎪ 2x + 3 y = 16)(−2) ⎩( Check: 21x + 6 y = 15 2(−1) + 3 y = 16 −4x − 6 y = −32 7(−1) + 2(6) = 5 −2 + 3 y = 16 +2 −7 + 12 = 5 17x = −17 +2 17 17 3 y = 18 2(−1) + 3(6) = 16 x = −1 3 3 −2 + 18 = 16 y=6 (−1, 6)Monday, September 19, 2011
  90. 90. Example 4 Solve each system of equations. Check your solution. ⎧x − 2 y = 7 ⎪ a. ⎨ ⎪−2x + 4 y = −14 ⎩Monday, September 19, 2011
  91. 91. Example 4 Solve each system of equations. Check your solution. ⎧x − 2 y = 7 ⎪ a. ⎨ ⎪−2x + 4 y = −14 ⎩ x = 2 y +7Monday, September 19, 2011
  92. 92. Example 4 Solve each system of equations. Check your solution. ⎧x − 2 y = 7 ⎪ a. ⎨ ⎪−2x + 4 y = −14 ⎩ x = 2 y +7 −2(2 y +7)+ 4 y = −14Monday, September 19, 2011
  93. 93. Example 4 Solve each system of equations. Check your solution. ⎧x − 2 y = 7 ⎪ a. ⎨ ⎪−2x + 4 y = −14 ⎩ x = 2 y +7 −2(2 y +7)+ 4 y = −14 −4 y −14 + 4 y = −14Monday, September 19, 2011
  94. 94. Example 4 Solve each system of equations. Check your solution. ⎧x − 2 y = 7 ⎪ a. ⎨ ⎪−2x + 4 y = −14 ⎩ x = 2 y +7 −2(2 y +7)+ 4 y = −14 −4 y −14 + 4 y = −14 −14 = −14Monday, September 19, 2011
  95. 95. Example 4 Solve each system of equations. Check your solution. ⎧x − 2 y = 7 ⎪ a. ⎨ ⎪−2x + 4 y = −14 ⎩ x = 2 y +7 −2(2 y +7)+ 4 y = −14 −4 y −14 + 4 y = −14 −14 = −14 What’s going on here?Monday, September 19, 2011
  96. 96. Example 4 Solve each system of equations. Check your solution. ⎧x − 2 y = 7 ⎪ a. ⎨ ⎪−2x + 4 y = −14 ⎩ x = 2 y +7 −2 y = −x +7 −2(2 y +7)+ 4 y = −14 −4 y −14 + 4 y = −14 −14 = −14 What’s going on here?Monday, September 19, 2011
  97. 97. Example 4 Solve each system of equations. Check your solution. ⎧x − 2 y = 7 ⎪ a. ⎨ ⎪−2x + 4 y = −14 ⎩ x = 2 y +7 −2 y = −x +7 −2(2 y +7)+ 4 y = −14 1 7 y= x− −4 y −14 + 4 y = −14 2 2 −14 = −14 What’s going on here?Monday, September 19, 2011
  98. 98. Example 4 Solve each system of equations. Check your solution. ⎧x − 2 y = 7 ⎪ a. ⎨ ⎪−2x + 4 y = −14 ⎩ x = 2 y +7 −2 y = −x +7 4 y = 2x −14 −2(2 y +7)+ 4 y = −14 1 7 y= x− −4 y −14 + 4 y = −14 2 2 −14 = −14 What’s going on here?Monday, September 19, 2011
  99. 99. Example 4 Solve each system of equations. Check your solution. ⎧x − 2 y = 7 ⎪ a. ⎨ ⎪−2x + 4 y = −14 ⎩ x = 2 y +7 −2 y = −x +7 4 y = 2x −14 −2(2 y +7)+ 4 y = −14 1 7 1 7 y= x− y= x− −4 y −14 + 4 y = −14 2 2 2 2 −14 = −14 What’s going on here?Monday, September 19, 2011
  100. 100. Example 4 Solve each system of equations. Check your solution. ⎧x − 2 y = 7 ⎪ a. ⎨ ⎪−2x + 4 y = −14 ⎩ x = 2 y +7 −2 y = −x +7 4 y = 2x −14 −2(2 y +7)+ 4 y = −14 1 7 1 7 y= x− y= x− −4 y −14 + 4 y = −14 2 2 2 2 −14 = −14 These are the same lines! What’s going on here?Monday, September 19, 2011
  101. 101. Example 4 Solve each system of equations. Check your solution. ⎧x − 2 y = 7 ⎪ a. ⎨ ⎪−2x + 4 y = −14 ⎩ x = 2 y +7 −2 y = −x +7 4 y = 2x −14 −2(2 y +7)+ 4 y = −14 1 7 1 7 y= x− y= x− −4 y −14 + 4 y = −14 2 2 2 2 −14 = −14 These are the same lines! What’s going on here? Infinitely many solutions on the line.Monday, September 19, 2011
  102. 102. Example 4 Solve each system of equations. Check your solution. ⎧x − 2 y = 7 ⎪ a. ⎨ ⎪−2x + 4 y = −14 ⎩ x = 2 y +7 −2 y = −x +7 4 y = 2x −14 −2(2 y +7)+ 4 y = −14 1 7 1 7 y= x− y= x− −4 y −14 + 4 y = −14 2 2 2 2 −14 = −14 These are the same lines! What’s going on here? Infinitely many solutions on the line.Monday, September 19, 2011
  103. 103. Example 4 Solve each system of equations. Check your solution. ⎧2x −7 y = −2 ⎪ b. ⎨ ⎪−4x +14 y = 3 ⎩Monday, September 19, 2011
  104. 104. Example 4 Solve each system of equations. Check your solution. ⎧2x −7 y = −2 ⎪ b. ⎨ ⎪−4x +14 y = 3 ⎩ −7 y = −2x − 2Monday, September 19, 2011
  105. 105. Example 4 Solve each system of equations. Check your solution. ⎧2x −7 y = −2 ⎪ b. ⎨ ⎪−4x +14 y = 3 ⎩ −7 y = −2x − 2 2 2 y= x− 7 7Monday, September 19, 2011
  106. 106. Example 4 Solve each system of equations. Check your solution. ⎧2x −7 y = −2 ⎪ b. ⎨ ⎪−4x +14 y = 3 ⎩ −7 y = −2x − 2 2 2 y= x− 7 7 ⎛2 2⎞ −4x +14 ⎜ x − ⎟ = 3 ⎝7 7⎠Monday, September 19, 2011
  107. 107. Example 4 Solve each system of equations. Check your solution. ⎧2x −7 y = −2 ⎪ b. ⎨ ⎪−4x +14 y = 3 ⎩ −7 y = −2x − 2 2 2 y= x− 7 7 ⎛2 2⎞ −4x +14 ⎜ x − ⎟ = 3 ⎝7 7⎠ −4x + 4x − 4 = 3Monday, September 19, 2011
  108. 108. Example 4 Solve each system of equations. Check your solution. ⎧2x −7 y = −2 ⎪ b. ⎨ ⎪−4x +14 y = 3 ⎩ −7 y = −2x − 2 2 2 y= x− 7 7 ⎛2 2⎞ −4x +14 ⎜ x − ⎟ = 3 ⎝7 7⎠ −4x + 4x − 4 = 3 −4 = 3Monday, September 19, 2011
  109. 109. Example 4 Solve each system of equations. Check your solution. ⎧2x −7 y = −2 ⎪ b. ⎨ ⎪−4x +14 y = 3 ⎩ −7 y = −2x − 2 2 2 14 y = 4x +3 y= x− 7 7 ⎛2 2⎞ −4x +14 ⎜ x − ⎟ = 3 ⎝7 7⎠ −4x + 4x − 4 = 3 −4 = 3Monday, September 19, 2011
  110. 110. Example 4 Solve each system of equations. Check your solution. ⎧2x −7 y = −2 ⎪ b. ⎨ ⎪−4x +14 y = 3 ⎩ −7 y = −2x − 2 2 2 14 y = 4x +3 y= x− 7 7 2 3 ⎛2 2⎞ y= x+ −4x +14 ⎜ x − ⎟ = 3 7 14 ⎝7 7⎠ −4x + 4x − 4 = 3 −4 = 3Monday, September 19, 2011
  111. 111. Example 4 Solve each system of equations. Check your solution. ⎧2x −7 y = −2 ⎪ b. ⎨ ⎪−4x +14 y = 3 ⎩ −7 y = −2x − 2 2 2 14 y = 4x +3 y= x− 7 7 2 3 ⎛2 2⎞ y= x+ −4x +14 ⎜ x − ⎟ = 3 7 14 ⎝7 7⎠ These lines are parallel. −4x + 4x − 4 = 3 −4 = 3Monday, September 19, 2011
  112. 112. Example 4 Solve each system of equations. Check your solution. ⎧2x −7 y = −2 ⎪ b. ⎨ ⎪−4x +14 y = 3 ⎩ −7 y = −2x − 2 2 2 14 y = 4x +3 y= x− 7 7 2 3 ⎛2 2⎞ y= x+ −4x +14 ⎜ x − ⎟ = 3 7 14 ⎝7 7⎠ These lines are parallel. −4x + 4x − 4 = 3 There are no solutions. −4 = 3Monday, September 19, 2011
  113. 113. When solving a system you get:Monday, September 19, 2011
  114. 114. When solving a system you get: One solution when:Monday, September 19, 2011
  115. 115. When solving a system you get: One solution when: No solutions when:Monday, September 19, 2011
  116. 116. When solving a system you get: One solution when: No solutions when: An infinite number of solutions on the line when:Monday, September 19, 2011
  117. 117. Problem SetMonday, September 19, 2011
  118. 118. Problem Set p. P18 #1-15 all “I have failed many times, and that’s why I am a success.” - Michael JordanMonday, September 19, 2011

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