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# AA Section 5-3

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Solving Systems by Substitution

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### AA Section 5-3

1. 1. 1. b 2. a 3. a. (6, 20); b. Plug into both equations 4. (2, 12) 5. (-8/3, -19,3) 6. b 7. It is a real world situation 8. a. Substitute to see answers are close; b. Substitute again 9. Advantage: Can see total number of solutions Disadvantage: Tough to ﬁnd exact answer 10. a. 0; b. n/a; c. n/a 12. a. 1; b. (2, 4); c. Plug in 14. a. Graph; b. 3 solutions; c. (-3.8, .2), (-1, 3), (.8, 4.8) 16. Graph the two equations; they do not intercept 18. 20. Whole numbers ≥ 21 -1 0 1 2 3 4 5 2 T − 2π r 24. x = 11 or x = -11 22. h = 2π r
2. 2. Section 5-3 Solving Systems by Substitution
3. 3. Warm-up 1. Solve 8x + 8(5-2x) = -40 2. Evaluate 3x - 2 when x = 4y + 1
4. 4. Warm-up 1. Solve 8x + 8(5-2x) = -40 8x + 8(5-2x) = -40 2. Evaluate 3x - 2 when x = 4y + 1
5. 5. Warm-up 1. Solve 8x + 8(5-2x) = -40 8x + 8(5-2x) = -40 8x + 40 -16x = -40 2. Evaluate 3x - 2 when x = 4y + 1
6. 6. Warm-up 1. Solve 8x + 8(5-2x) = -40 8x + 8(5-2x) = -40 8x + 40 -16x = -40 -8x + 40 = -40 2. Evaluate 3x - 2 when x = 4y + 1
7. 7. Warm-up 1. Solve 8x + 8(5-2x) = -40 8x + 8(5-2x) = -40 8x + 40 -16x = -40 -8x + 40 = -40 -8x = -80 2. Evaluate 3x - 2 when x = 4y + 1
8. 8. Warm-up 1. Solve 8x + 8(5-2x) = -40 8x + 8(5-2x) = -40 8x + 40 -16x = -40 -8x + 40 = -40 -8x = -80 x = 10 2. Evaluate 3x - 2 when x = 4y + 1
9. 9. Warm-up 1. Solve 8x + 8(5-2x) = -40 8x + 8(5-2x) = -40 8x + 40 -16x = -40 -8x + 40 = -40 -8x = -80 x = 10 2. Evaluate 3x - 2 when x = 4y + 1 3(4y + 1) - 2
10. 10. Warm-up 1. Solve 8x + 8(5-2x) = -40 8x + 8(5-2x) = -40 8x + 40 -16x = -40 -8x + 40 = -40 -8x = -80 x = 10 2. Evaluate 3x - 2 when x = 4y + 1 3(4y + 1) - 2 12y + 3 - 2
11. 11. Warm-up 1. Solve 8x + 8(5-2x) = -40 8x + 8(5-2x) = -40 8x + 40 -16x = -40 -8x + 40 = -40 -8x = -80 x = 10 2. Evaluate 3x - 2 when x = 4y + 1 3(4y + 1) - 2 12y + 3 - 2 12y + 1
12. 12. 1. Tables
13. 13. 1. Tables Not very efﬁcient
14. 14. 1. Tables Not very efﬁcient 2. Graphing by hand
15. 15. 1. Tables Not very efﬁcient 2. Graphing by hand Not very accurate
16. 16. 1. Tables Not very efﬁcient 2. Graphing by hand Not very accurate 3. Graphing Calculator
17. 17. 1. Tables Not very efﬁcient 2. Graphing by hand Not very accurate 3. Graphing Calculator Cheap way out
18. 18. Example 1 Solve. x + y = 6  y = x + 2 
19. 19. Example 1 Solve. x + y = 6  y = x + 2  x + (x + 2) = 6
20. 20. Example 1 Solve. x + y = 6  y = x + 2  x + (x + 2) = 6 2x + 2 = 6
21. 21. Example 1 Solve. x + y = 6  y = x + 2  x + (x + 2) = 6 2x + 2 = 6 2x = 4
22. 22. Example 1 Solve. x + y = 6  y = x + 2  x + (x + 2) = 6 2x + 2 = 6 2x = 4 x=2
23. 23. Example 1 Solve. x + y = 6  y = x + 2  x + (x + 2) = 6 y=x+2 2x + 2 = 6 2x = 4 x=2
24. 24. Example 1 Solve. x + y = 6  y = x + 2  x + (x + 2) = 6 y=x+2 2x + 2 = 6 y=2+2 2x = 4 x=2
25. 25. Example 1 Solve. x + y = 6  y = x + 2  x + (x + 2) = 6 y=x+2 2x + 2 = 6 y=2+2 2x = 4 y=4 x=2
26. 26. Example 1 Solve. x + y = 6  y = x + 2  x + (x + 2) = 6 y=x+2 x+y=6 2x + 2 = 6 y=2+2 2x = 4 y=4 x=2
27. 27. Example 1 Solve. x + y = 6  y = x + 2  x + (x + 2) = 6 y=x+2 x+y=6 2x + 2 = 6 y=2+2 2+4=6 2x = 4 y=4 x=2
28. 28. Example 1 Solve. x + y = 6  y = x + 2  x + (x + 2) = 6 y=x+2 x+y=6 2x + 2 = 6 y=2+2 2+4=6 2x = 4 y=4 x=2 (2, 4)
29. 29. Example 1 Solve. x + y = 6  y = x + 2  x + (x + 2) = 6 y=x+2 x+y=6 2x + 2 = 6 y=2+2 2+4=6 2x = 4 y=4 x=2 (2, 4) Always check your answer.
30. 30. Example 1 Solve. x + y = 6  y = x + 2  x + (x + 2) = 6 y=x+2 x+y=6 2x + 2 = 6 y=2+2 2+4=6 2x = 4 y=4 x=2 (2, 4) Always check your answer. You’ll know you’re right.
31. 31. Example 2 The Drama Club printed 1750 tickets for their spring play. They printed twice as many student tickets as adult tickets and half as many children’s tickets as adult tickets. Write a system of 3 equations and ﬁnd the number of each ticket printed.
32. 32. Example 2 The Drama Club printed 1750 tickets for their spring play. They printed twice as many student tickets as adult tickets and half as many children’s tickets as adult tickets. Write a system of 3 equations and ﬁnd the number of each ticket printed. A = adult tickets S = student tickets C = children tickets
33. 33. Example 2 The Drama Club printed 1750 tickets for their spring play. They printed twice as many student tickets as adult tickets and half as many children’s tickets as adult tickets. Write a system of 3 equations and ﬁnd the number of each ticket printed. A = adult tickets S = student tickets C = children tickets     
34. 34. Example 2 The Drama Club printed 1750 tickets for their spring play. They printed twice as many student tickets as adult tickets and half as many children’s tickets as adult tickets. Write a system of 3 equations and ﬁnd the number of each ticket printed. A = adult tickets S = student tickets C = children tickets  A + S + C = 1750    
35. 35. Example 2 The Drama Club printed 1750 tickets for their spring play. They printed twice as many student tickets as adult tickets and half as many children’s tickets as adult tickets. Write a system of 3 equations and ﬁnd the number of each ticket printed. A = adult tickets S = student tickets C = children tickets  A + S + C = 1750   S = 2A  
36. 36. Example 2 The Drama Club printed 1750 tickets for their spring play. They printed twice as many student tickets as adult tickets and half as many children’s tickets as adult tickets. Write a system of 3 equations and ﬁnd the number of each ticket printed. A = adult tickets S = student tickets C = children tickets  A + S + C = 1750   S = 2A   C = 1/2 A
37. 37. Example 2 The Drama Club printed 1750 tickets for their spring play. They printed twice as many student tickets as adult tickets and half as many children’s tickets as adult tickets. Write a system of 3 equations and ﬁnd the number of each ticket printed. A = adult tickets S = student tickets C = children tickets  A + S + C = 1750 A + 2A + 1/2 A = 1750   S = 2A   C = 1/2 A
38. 38. Example 2 The Drama Club printed 1750 tickets for their spring play. They printed twice as many student tickets as adult tickets and half as many children’s tickets as adult tickets. Write a system of 3 equations and ﬁnd the number of each ticket printed. A = adult tickets S = student tickets C = children tickets  A + S + C = 1750 A + 2A + 1/2 A = 1750   S = 2A 7/2 A = 1750   C = 1/2 A
39. 39. Example 2 The Drama Club printed 1750 tickets for their spring play. They printed twice as many student tickets as adult tickets and half as many children’s tickets as adult tickets. Write a system of 3 equations and ﬁnd the number of each ticket printed. A = adult tickets S = student tickets C = children tickets  A + S + C = 1750 A + 2A + 1/2 A = 1750   S = 2A 7/2 A = 1750   C = 1/2 A A = 500
40. 40. Example 2 The Drama Club printed 1750 tickets for their spring play. They printed twice as many student tickets as adult tickets and half as many children’s tickets as adult tickets. Write a system of 3 equations and ﬁnd the number of each ticket printed. A = adult tickets S = student tickets C = children tickets  A + S + C = 1750 A + 2A + 1/2 A = 1750   S = 2A 7/2 A = 1750   C = 1/2 A A = 500 S = 1000
41. 41. Example 2 The Drama Club printed 1750 tickets for their spring play. They printed twice as many student tickets as adult tickets and half as many children’s tickets as adult tickets. Write a system of 3 equations and ﬁnd the number of each ticket printed. A = adult tickets S = student tickets C = children tickets  A + S + C = 1750 A + 2A + 1/2 A = 1750   S = 2A 7/2 A = 1750   C = 1/2 A A = 500 S = 1000 C = 250
42. 42. Example 2 The Drama Club printed 1750 tickets for their spring play. They printed twice as many student tickets as adult tickets and half as many children’s tickets as adult tickets. Write a system of 3 equations and ﬁnd the number of each ticket printed. A = adult tickets S = student tickets C = children tickets  A + S + C = 1750 A + 2A + 1/2 A = 1750   S = 2A 7/2 A = 1750   C = 1/2 A A = 500 S = 1000 They printed 500 adult tickets, 1000 student C = 250 tickets, and 250 children’s tickets
43. 43. Example 3 Solve. y = 4x   xy = 36 
44. 44. Example 3 Solve. y = 4x   xy = 36  x(4x) = 36
45. 45. Example 3 Solve. y = 4x   xy = 36  x(4x) = 36 4x2 = 36
46. 46. Example 3 Solve. y = 4x   xy = 36  x(4x) = 36 4x2 = 36 x2 = 9
47. 47. Example 3 Solve. y = 4x   xy = 36  x(4x) = 36 4x2 = 36 x2 = 9 x2 = ± 9
48. 48. Example 3 Solve. y = 4x   xy = 36  x(4x) = 36 4x2 = 36 x2 = 9 x2 = ± 9 x = 3 or x = -3
49. 49. Example 3 Solve. y = 4x   xy = 36  x(4x) = 36 y = 4(3) 4x2 = 36 x2 = 9 x2 = ± 9 x = 3 or x = -3
50. 50. Example 3 Solve. y = 4x   xy = 36  x(4x) = 36 y = 4(3) 4x2 = 36 y = 12 x2 = 9 x2 = ± 9 x = 3 or x = -3
51. 51. Example 3 Solve. y = 4x   xy = 36  x(4x) = 36 y = 4(3) 4x2 = 36 y = 12 x2 = 9 y = 4(-3) x2 = ± 9 x = 3 or x = -3
52. 52. Example 3 Solve. y = 4x   xy = 36  x(4x) = 36 y = 4(3) 4x2 = 36 y = 12 x2 = 9 y = 4(-3) x2 = ± 9 y = -12 x = 3 or x = -3
53. 53. Example 3 Solve. y = 4x   xy = 36  x(4x) = 36 y = 4(3) Check: 4x2 = 36 y = 12 x2 = 9 y = 4(-3) x2 = ± 9 y = -12 x = 3 or x = -3
54. 54. Example 3 Solve. y = 4x   xy = 36  x(4x) = 36 y = 4(3) Check: 4x2 = 36 y = 12 (3)(12) = 36 x2 = 9 y = 4(-3) x2 = ± 9 y = -12 x = 3 or x = -3
55. 55. Example 3 Solve. y = 4x   xy = 36  x(4x) = 36 y = 4(3) Check: 4x2 = 36 y = 12 (3)(12) = 36 x2 = 9 y = 4(-3) (-3)(-12) = 36 x2 = ± 9 y = -12 x = 3 or x = -3
56. 56. Example 3 Solve. y = 4x   xy = 36  x(4x) = 36 y = 4(3) Check: 4x2 = 36 y = 12 (3)(12) = 36 x2 = 9 y = 4(-3) (-3)(-12) = 36 x2 = ± 9 y = -12 x = 3 or x = -3 (3, 12) or (-3, -12)
57. 57. Example 4 Solve. y = 4 − 3x  3x + y = 7 
58. 58. Example 4 Solve. y = 4 − 3x  3x + y = 7  3x + (4 - 3x) = 7
59. 59. Example 4 Solve. y = 4 − 3x  3x + y = 7  3x + (4 - 3x) = 7 4=7
60. 60. Example 4 Solve. y = 4 − 3x  3x + y = 7  3x + (4 - 3x) = 7 4≠7
61. 61. Example 4 Solve. y = 4 − 3x  3x + y = 7  3x + (4 - 3x) = 7 4≠7 Wait, what?
62. 62. Example 4 Solve. y = 4 − 3x  3x + y = 7  3x + (4 - 3x) = 7 4≠7 Wait, what? 3x + y = 7
63. 63. Example 4 Solve. y = 4 − 3x  3x + y = 7  3x + (4 - 3x) = 7 4≠7 Wait, what? 3x + y = 7 y = -3x + 7
64. 64. Example 4 Solve. y = 4 − 3x  3x + y = 7  3x + (4 - 3x) = 7 4≠7 Wait, what? 3x + y = 7 y = -3x + 7 Oh, parallel lines!
65. 65. Example 4 Solve. y = 4 − 3x  3x + y = 7  3x + (4 - 3x) = 7 4≠7 Wait, what? 3x + y = 7 y = -3x + 7 Oh, parallel lines! (No solutions)
66. 66. Example 5 Solve. y = 2x 2   2 3y = 6x 
67. 67. Example 5 Solve. y = 2x 2   2 3y = 6x  2 2 3(2x ) = 6x
68. 68. Example 5 Solve. y = 2x 2   2 3y = 6x  2 2 3(2x ) = 6x 2 2 6x = 6x
69. 69. Example 5 Solve. y = 2x 2   2 3y = 6x  2 2 3(2x ) = 6x 2 2 6x = 6x This is always true!
70. 70. Example 5 Solve. y = 2x 2   2 3y = 6x  2 2 3(2x ) = 6x 2 2 6x = 6x This is always true! These are the same graphs.
71. 71. Example 5 Solve. y = 2x 2   2 3y = 6x  2 2 3(2x ) = 6x 2 2 6x = 6x This is always true! These are the same graphs. Inﬁnitely many solutions on the parabola
72. 72. Consistent:
73. 73. Consistent: A system with one or more solutions
74. 74. Consistent: A system with one or more solutions Inconsistent:
75. 75. Consistent: A system with one or more solutions Inconsistent: A systems with no solutions
76. 76. Homework
77. 77. Homework p. 289 #1-20, skip 17, 18 “Too many people are thinking of security instead of opportunity. They seem more afraid of life than death.” - James F. Byrnes