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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