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# Integrated Math 2 Section 8-4

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Solve Systems by Adding, Subtracting, and Multiplying

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• ### Integrated Math 2 Section 8-4

1. 1. Section 8-4 Solve Systems by Adding, Subtracting, and Multiplying (Linear Combinations)
2. 2. Essential Questions How do you solve systems of equations by adding and subtracting? How do you solve systems of equations by adding and multiplying? Where you’ll see this: Landscaping, construction, sports
3. 3. Why another method?
4. 4. Why another method? Graphing: Problems with accuracy
5. 5. Why another method? Graphing: Problems with accuracy Substitution: Need an isolated variable
6. 6. Why another method? Graphing: Problems with accuracy Substitution: Need an isolated variable Combinations: Could speed up process
7. 7. Why another method? Graphing: Problems with accuracy Substitution: Need an isolated variable Combinations: Could speed up process Look at the system to determine best method
8. 8. Solve by Combinations
9. 9. Solve by Combinations 1. Choose a variable to eliminate (your choice).
10. 10. Solve by Combinations 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.
11. 11. Solve by Combinations 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.
12. 12. Solve by Combinations 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.
13. 13. Solve by Combinations 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.
14. 14. Example 1 Solve by combining the equations −2x + 3 y = 2  a.   2x + 7 y = 18 
15. 15. Example 1 Solve by combining the equations −2x + 3 y = 2  a.   2x + 7 y = 18  −2x + 3 y = 2
16. 16. Example 1 Solve by combining the equations −2x + 3 y = 2  a.   2x + 7 y = 18  −2x + 3 y = 2 2x + 7 y = 18
17. 17. Example 1 Solve by combining the equations −2x + 3 y = 2  a.   2x + 7 y = 18  −2x + 3 y = 2 2x + 7 y = 18
18. 18. Example 1 Solve by combining the equations −2x + 3 y = 2  a.   2x + 7 y = 18  −2x + 3 y = 2 2x + 7 y = 18 10 y = 20
19. 19. Example 1 Solve by combining the equations −2x + 3 y = 2  a.   2x + 7 y = 18  −2x + 3 y = 2 2x + 7 y = 18 10 y = 20 10 10
20. 20. Example 1 Solve by combining the equations −2x + 3 y = 2  a.   2x + 7 y = 18  −2x + 3 y = 2 2x + 7 y = 18 10 y = 20 10 10 y=2
21. 21. Example 1 Solve by combining the equations −2x + 3 y = 2  a.   2x + 7 y = 18  −2x + 3 y = 2 −2x + 3(2) = 2 2x + 7 y = 18 10 y = 20 10 10 y=2
22. 22. Example 1 Solve by combining the equations −2x + 3 y = 2  a.   2x + 7 y = 18  −2x + 3 y = 2 −2x + 3(2) = 2 2x + 7 y = 18 −2x + 6 = 2 10 y = 20 10 10 y=2
23. 23. Example 1 Solve by combining the equations −2x + 3 y = 2  a.   2x + 7 y = 18  −2x + 3 y = 2 −2x + 3(2) = 2 2x + 7 y = 18 −2x + 6 = 2 10 y = 20 −6 −6 10 10 y=2
24. 24. Example 1 Solve by combining the equations −2x + 3 y = 2  a.   2x + 7 y = 18  −2x + 3 y = 2 −2x + 3(2) = 2 2x + 7 y = 18 −2x + 6 = 2 10 y = 20 −6 −6 10 10 −2x = −4 y=2
25. 25. Example 1 Solve by combining the equations −2x + 3 y = 2  a.   2x + 7 y = 18  −2x + 3 y = 2 −2x + 3(2) = 2 2x + 7 y = 18 −2x + 6 = 2 10 y = 20 −6 −6 10 10 −2x = −4 y=2 −2 −2
26. 26. Example 1 Solve by combining the equations −2x + 3 y = 2  a.   2x + 7 y = 18  −2x + 3 y = 2 −2x + 3(2) = 2 2x + 7 y = 18 −2x + 6 = 2 10 y = 20 −6 −6 10 10 −2x = −4 y=2 −2 −2 x=2
27. 27. Example 1 Solve by combining the equations −2x + 3 y = 2  a.   2x + 7 y = 18  Check: −2x + 3 y = 2 −2x + 3(2) = 2 2x + 7 y = 18 −2x + 6 = 2 10 y = 20 −6 −6 10 10 −2x = −4 y=2 −2 −2 x=2
28. 28. Example 1 Solve by combining the equations −2x + 3 y = 2  a.   2x + 7 y = 18  Check: −2x + 3 y = 2 −2(2) + 3(2) = 2 −2x + 3(2) = 2 2x + 7 y = 18 −2x + 6 = 2 10 y = 20 −6 −6 10 10 −2x = −4 y=2 −2 −2 x=2
29. 29. Example 1 Solve by combining the equations −2x + 3 y = 2  a.   2x + 7 y = 18  Check: −2x + 3 y = 2 −2(2) + 3(2) = 2 −2x + 3(2) = 2 2x + 7 y = 18 −4 + 6 = 2 −2x + 6 = 2 10 y = 20 −6 −6 10 10 −2x = −4 y=2 −2 −2 x=2
30. 30. Example 1 Solve by combining the equations −2x + 3 y = 2  a.   2x + 7 y = 18  Check: −2x + 3 y = 2 −2(2) + 3(2) = 2 −2x + 3(2) = 2 2x + 7 y = 18 −4 + 6 = 2 −2x + 6 = 2 10 y = 20 −6 −6 2(2) + 7(2) = 18 10 10 −2x = −4 y=2 −2 −2 x=2
31. 31. Example 1 Solve by combining the equations −2x + 3 y = 2  a.   2x + 7 y = 18  Check: −2x + 3 y = 2 −2(2) + 3(2) = 2 −2x + 3(2) = 2 2x + 7 y = 18 −4 + 6 = 2 −2x + 6 = 2 10 y = 20 −6 −6 2(2) + 7(2) = 18 10 10 −2x = −4 4 + 14 = 18 y=2 −2 −2 x=2
32. 32. Example 1 Solve by combining the equations −2x + 3 y = 2  a.   2x + 7 y = 18  Check: −2x + 3 y = 2 −2(2) + 3(2) = 2 −2x + 3(2) = 2 2x + 7 y = 18 −4 + 6 = 2 −2x + 6 = 2 10 y = 20 −6 −6 2(2) + 7(2) = 18 10 10 −2x = −4 4 + 14 = 18 y=2 −2 −2 x=2 (2, 2)
33. 33. Example 1 Solve by combining the equations 10x + 4 y = 14  b.  x + 4 y = 5 
34. 34. Example 1 Solve by combining the equations 10x + 4 y = 14  b.   x + 4 y = 5)(−1) (
35. 35. Example 1 Solve by combining the equations 10x + 4 y = 14  b.   x + 4 y = 5)(−1) ( 10x + 4 y = 14
36. 36. Example 1 Solve by combining the equations 10x + 4 y = 14  b.   x + 4 y = 5)(−1) ( 10x + 4 y = 14 − x − 4 y = −5
37. 37. Example 1 Solve by combining the equations 10x + 4 y = 14  b.   x + 4 y = 5)(−1) ( 10x + 4 y = 14 − x − 4 y = −5
38. 38. Example 1 Solve by combining the equations 10x + 4 y = 14  b.   x + 4 y = 5)(−1) ( 10x + 4 y = 14 − x − 4 y = −5 9x = 9
39. 39. Example 1 Solve by combining the equations 10x + 4 y = 14  b.   x + 4 y = 5)(−1) ( 10x + 4 y = 14 − x − 4 y = −5 9x = 9 9 9
40. 40. Example 1 Solve by combining the equations 10x + 4 y = 14  b.   x + 4 y = 5)(−1) ( 10x + 4 y = 14 − x − 4 y = −5 9x = 9 9 9 x=1
41. 41. Example 1 Solve by combining the equations 10x + 4 y = 14  b.   x + 4 y = 5)(−1) ( 10x + 4 y = 14 − x − 4 y = −5 1+ 4 y = 5 9x = 9 9 9 x=1
42. 42. Example 1 Solve by combining the equations 10x + 4 y = 14  b.   x + 4 y = 5)(−1) ( 10x + 4 y = 14 − x − 4 y = −5 1+ 4 y = 5 9x = 9 −1 −1 9 9 x=1
43. 43. Example 1 Solve by combining the equations 10x + 4 y = 14  b.   x + 4 y = 5)(−1) ( 10x + 4 y = 14 − x − 4 y = −5 1+ 4 y = 5 9x = 9 −1 −1 9 9 4y = 4 x=1
44. 44. Example 1 Solve by combining the equations 10x + 4 y = 14  b.   x + 4 y = 5)(−1) ( 10x + 4 y = 14 − x − 4 y = −5 1+ 4 y = 5 9x = 9 −1 −1 9 9 4y = 4 4 4 x=1
45. 45. Example 1 Solve by combining the equations 10x + 4 y = 14  b.   x + 4 y = 5)(−1) ( 10x + 4 y = 14 − x − 4 y = −5 1+ 4 y = 5 9x = 9 −1 −1 9 9 4y = 4 4 4 x=1 y =1
46. 46. Example 1 Solve by combining the equations 10x + 4 y = 14  b.   x + 4 y = 5)(−1) ( Check: 10x + 4 y = 14 − x − 4 y = −5 1+ 4 y = 5 9x = 9 −1 −1 9 9 4y = 4 4 4 x=1 y =1
47. 47. Example 1 Solve by combining the equations 10x + 4 y = 14  b.   x + 4 y = 5)(−1) ( Check: 10x + 4 y = 14 − x − 4 y = −5 1+ 4 y = 5 10(1) + 4(1) = 14 9x = 9 −1 −1 9 9 4y = 4 4 4 x=1 y =1
48. 48. Example 1 Solve by combining the equations 10x + 4 y = 14  b.   x + 4 y = 5)(−1) ( Check: 10x + 4 y = 14 − x − 4 y = −5 1+ 4 y = 5 10(1) + 4(1) = 14 9x = 9 −1 −1 10 + 4 = 14 9 9 4y = 4 4 4 x=1 y =1
49. 49. Example 1 Solve by combining the equations 10x + 4 y = 14  b.   x + 4 y = 5)(−1) ( Check: 10x + 4 y = 14 − x − 4 y = −5 1+ 4 y = 5 10(1) + 4(1) = 14 9x = 9 −1 −1 10 + 4 = 14 9 9 4y = 4 4 4 1 + 4(1) = 5 x=1 y =1
50. 50. Example 1 Solve by combining the equations 10x + 4 y = 14  b.   x + 4 y = 5)(−1) ( Check: 10x + 4 y = 14 − x − 4 y = −5 1+ 4 y = 5 10(1) + 4(1) = 14 9x = 9 −1 −1 10 + 4 = 14 9 9 4y = 4 4 4 1 + 4(1) = 5 x=1 1+ 4 = 5 y =1
51. 51. Example 1 Solve by combining the equations 10x + 4 y = 14  b.   x + 4 y = 5)(−1) ( Check: 10x + 4 y = 14 − x − 4 y = −5 1+ 4 y = 5 10(1) + 4(1) = 14 9x = 9 −1 −1 10 + 4 = 14 9 9 4y = 4 4 4 1 + 4(1) = 5 x=1 1+ 4 = 5 y =1 (1,1)
52. 52. Example 2 Solve by combining the equations 7x + 2 y = 5    2x + 3 y = 16 
53. 53. Example 2 Solve by combining the equations 7x + 2 y = 5)(3) (   2x + 3 y = 16 
54. 54. Example 2 Solve by combining the equations 7x + 2 y = 5)(3) (   2x + 3 y = 16)(−2) (
55. 55. Example 2 Solve by combining the equations 7x + 2 y = 5)(3) (   2x + 3 y = 16)(−2) ( 21x + 6 y = 15
56. 56. Example 2 Solve by combining the equations 7x + 2 y = 5)(3) (   2x + 3 y = 16)(−2) ( 21x + 6 y = 15 −4x − 6 y = −32
57. 57. Example 2 Solve by combining the equations 7x + 2 y = 5)(3) (   2x + 3 y = 16)(−2) ( 21x + 6 y = 15 −4x − 6 y = −32
58. 58. Example 2 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
59. 59. Example 2 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
60. 60. Example 2 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 = −1
61. 61. Example 2 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 = −1
62. 62. Example 2 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 = −1
63. 63. Example 2 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 = −1
64. 64. Example 2 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
65. 65. Example 2 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
66. 66. Example 2 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=6
67. 67. Example 2 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=6
68. 68. Example 2 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=6
69. 69. Example 2 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 −7 + 12 = 5 17x = −17 +2 +2 17 17 3 y = 18 x = −1 3 3 y=6
70. 70. Example 2 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 −7 + 12 = 5 17x = −17 +2 +2 17 17 3 y = 18 2(−1) + 3(6) = 16 x = −1 3 3 y=6
71. 71. Example 2 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 −7 + 12 = 5 17x = −17 +2 +2 17 17 3 y = 18 2(−1) + 3(6) = 16 x = −1 3 3 −2 + 18 = 16 y=6
72. 72. Example 2 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 −7 + 12 = 5 17x = −17 +2 +2 17 17 3 y = 18 2(−1) + 3(6) = 16 x = −1 3 3 −2 + 18 = 16 y=6 (−1, 6)
73. 73. Example 3 Solve by combining the equations 5x + 2 y = 7   4x + 2 y = −3 
74. 74. Example 3 Solve by combining the equations 5x + 2 y = 7   4x + 2 y = −3)(−1) (
75. 75. Example 3 Solve by combining the equations 5x + 2 y = 7   4x + 2 y = −3)(−1) ( 5x + 2 y = 7
76. 76. Example 3 Solve by combining the equations 5x + 2 y = 7   4x + 2 y = −3)(−1) ( 5x + 2 y = 7 −4x − 2 y = 3
77. 77. Example 3 Solve by combining the equations 5x + 2 y = 7   4x + 2 y = −3)(−1) ( 5x + 2 y = 7 −4x − 2 y = 3
78. 78. Example 3 Solve by combining the equations 5x + 2 y = 7   4x + 2 y = −3)(−1) ( 5x + 2 y = 7 −4x − 2 y = 3 x = 10
79. 79. Example 3 Solve by combining the equations 5x + 2 y = 7   4x + 2 y = −3)(−1) ( 5x + 2 y = 7 5(10) + 2 y = 7 −4x − 2 y = 3 x = 10
80. 80. Example 3 Solve by combining the equations 5x + 2 y = 7   4x + 2 y = −3)(−1) ( 5x + 2 y = 7 5(10) + 2 y = 7 −4x − 2 y = 3 50 + 2 y = 7 x = 10
81. 81. Example 3 Solve by combining the equations 5x + 2 y = 7   4x + 2 y = −3)(−1) ( 5x + 2 y = 7 5(10) + 2 y = 7 −4x − 2 y = 3 50 + 2 y = 7 −50 −50 x = 10
82. 82. Example 3 Solve by combining the equations 5x + 2 y = 7   4x + 2 y = −3)(−1) ( 5x + 2 y = 7 5(10) + 2 y = 7 −4x − 2 y = 3 50 + 2 y = 7 −50 −50 x = 10 2 y = −43
83. 83. Example 3 Solve by combining the equations 5x + 2 y = 7   4x + 2 y = −3)(−1) ( 5x + 2 y = 7 5(10) + 2 y = 7 −4x − 2 y = 3 50 + 2 y = 7 −50 −50 x = 10 2 y = −43 2 2
84. 84. Example 3 Solve by combining the equations 5x + 2 y = 7   4x + 2 y = −3)(−1) ( 5x + 2 y = 7 5(10) + 2 y = 7 −4x − 2 y = 3 50 + 2 y = 7 −50 −50 x = 10 2 y = −43 2 2 −43 y= 2
85. 85. Example 3 Solve by combining the equations 5x + 2 y = 7   4x + 2 y = −3)(−1) ( Check: 5x + 2 y = 7 5(10) + 2 y = 7 −4x − 2 y = 3 50 + 2 y = 7 −50 −50 x = 10 2 y = −43 2 2 −43 y= 2
86. 86. Example 3 Solve by combining the equations 5x + 2 y = 7   4x + 2 y = −3)(−1) ( Check: 5x + 2 y = 7 5(10) + 2 y = 7 5(10) + 2(− ) = 7 43 −4x − 2 y = 3 50 + 2 y = 7 2 −50 −50 x = 10 2 y = −43 2 2 −43 y= 2
87. 87. Example 3 Solve by combining the equations 5x + 2 y = 7   4x + 2 y = −3)(−1) ( Check: 5x + 2 y = 7 5(10) + 2 y = 7 5(10) + 2(− ) = 7 43 −4x − 2 y = 3 50 + 2 y = 7 2 −50 50 − 43 = 7 −50 x = 10 2 y = −43 2 2 −43 y= 2
88. 88. Example 3 Solve by combining the equations 5x + 2 y = 7   4x + 2 y = −3)(−1) ( Check: 5x + 2 y = 7 5(10) + 2 y = 7 5(10) + 2(− ) = 7 43 −4x − 2 y = 3 50 + 2 y = 7 2 −50 50 − 43 = 7 −50 x = 10 2 y = −43 4(10) + 2(− ) = −3 43 2 2 2 −43 y= 2
89. 89. Example 3 Solve by combining the equations 5x + 2 y = 7   4x + 2 y = −3)(−1) ( Check: 5x + 2 y = 7 5(10) + 2 y = 7 5(10) + 2(− ) = 7 43 −4x − 2 y = 3 50 + 2 y = 7 2 −50 50 − 43 = 7 −50 x = 10 2 y = −43 4(10) + 2(− ) = −3 43 2 2 2 40 − 43 = −3 −43 y= 2
90. 90. Example 3 Solve by combining the equations 5x + 2 y = 7   4x + 2 y = −3)(−1) ( Check: 5x + 2 y = 7 5(10) + 2 y = 7 5(10) + 2(− ) = 7 43 −4x − 2 y = 3 50 + 2 y = 7 2 −50 50 − 43 = 7 −50 x = 10 2 y = −43 4(10) + 2(− ) = −3 43 2 2 2 40 − 43 = −3 −43 y= 2 (10, − ) 43 2
91. 91. Homework
92. 92. Homework p. 350 #1-27 odd “A verbal contract isn’t worth the paper it’s written on.” - Samuel Goldwyn