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Int Math 2 Section 3-1 1011

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Equations and Formulas

Equations and Formulas

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  • 1. Chapter 3 Equations and Inequalities
  • 2. Section 3-1 Equations and Formulas
  • 3. Essential Questions ✤ How do you determine if a number is a solution of an equation? ✤ How do you solve an equation or a formula? ✤ Where you’ll see it: ✤ Travel, safety, physics
  • 4. Vocabulary 1. Equation: 2. True Equation: 3. False Equation: 4. Open Sentence: 5. Solution of the Equation: 6. Solve an Equation:
  • 5. Vocabulary 1. Equation: A statement where two numbers or expressions are equal 2. True Equation: 3. False Equation: 4. Open Sentence: 5. Solution of the Equation: 6. Solve an Equation:
  • 6. Vocabulary 1. Equation: A statement where two numbers or expressions are equal 2. True Equation: Both sides have the same numerical expression 3. False Equation: 4. Open Sentence: 5. Solution of the Equation: 6. Solve an Equation:
  • 7. Vocabulary 1. Equation: A statement where two numbers or expressions are equal 2. True Equation: Both sides have the same numerical expression 3. False Equation: When the numerical expressions are not equal 4. Open Sentence: 5. Solution of the Equation: 6. Solve an Equation:
  • 8. Vocabulary 1. Equation: A statement where two numbers or expressions are equal 2. True Equation: Both sides have the same numerical expression 3. False Equation: When the numerical expressions are not equal 4. Open Sentence: An equation with a variable 5. Solution of the Equation: 6. Solve an Equation:
  • 9. Vocabulary 1. Equation: A statement where two numbers or expressions are equal 2. True Equation: Both sides have the same numerical expression 3. False Equation: When the numerical expressions are not equal 4. Open Sentence: An equation with a variable 5. Solution of the Equation: The value that makes the equation true 6. Solve an Equation:
  • 10. Vocabulary 1. Equation: A statement where two numbers or expressions are equal 2. True Equation: Both sides have the same numerical expression 3. False Equation: When the numerical expressions are not equal 4. Open Sentence: An equation with a variable 5. Solution of the Equation: The value that makes the equation true 6. Solve an Equation: Find the values that make it true by isolating the variable
  • 11. Formula
  • 12. Formula An equation (rule) that states a relationship between two or more quantities
  • 13. Formula An equation (rule) that states a relationship between two or more quantities We can solve for any part of a formula!!!
  • 14. Example 1 Using only the numbers from {-3, -2, -1, 0, 1, 2, 3}, determine the solution(s) of each equation. 2 a. 14x + 3 = 12x + 7 b. x = 9 c. 3x + 8 = 5
  • 15. Example 1 Using only the numbers from {-3, -2, -1, 0, 1, 2, 3}, determine the solution(s) of each equation. 2 a. 14x + 3 = 12x + 7 b. x = 9 c. 3x + 8 = 5 −12x
  • 16. Example 1 Using only the numbers from {-3, -2, -1, 0, 1, 2, 3}, determine the solution(s) of each equation. 2 a. 14x + 3 = 12x + 7 b. x = 9 c. 3x + 8 = 5 −12x −12x
  • 17. Example 1 Using only the numbers from {-3, -2, -1, 0, 1, 2, 3}, determine the solution(s) of each equation. 2 a. 14x + 3 = 12x + 7 b. x = 9 c. 3x + 8 = 5 −12x −12x
  • 18. Example 1 Using only the numbers from {-3, -2, -1, 0, 1, 2, 3}, determine the solution(s) of each equation. 2 a. 14x + 3 = 12x + 7 b. x = 9 c. 3x + 8 = 5 −12x −12x 2x + 3 = 7
  • 19. Example 1 Using only the numbers from {-3, -2, -1, 0, 1, 2, 3}, determine the solution(s) of each equation. 2 a. 14x + 3 = 12x + 7 b. x = 9 c. 3x + 8 = 5 −12x −12x 2x + 3 = 7 −3 −3
  • 20. Example 1 Using only the numbers from {-3, -2, -1, 0, 1, 2, 3}, determine the solution(s) of each equation. 2 a. 14x + 3 = 12x + 7 b. x = 9 c. 3x + 8 = 5 −12x −12x 2x + 3 = 7 −3 −3 2x = 4
  • 21. Example 1 Using only the numbers from {-3, -2, -1, 0, 1, 2, 3}, determine the solution(s) of each equation. 2 a. 14x + 3 = 12x + 7 b. x = 9 c. 3x + 8 = 5 −12x −12x 2x + 3 = 7 −3 −3 2x = 4 2 2
  • 22. Example 1 Using only the numbers from {-3, -2, -1, 0, 1, 2, 3}, determine the solution(s) of each equation. 2 a. 14x + 3 = 12x + 7 b. x = 9 c. 3x + 8 = 5 −12x −12x 2x + 3 = 7 −3 −3 2x = 4 2 2 x=2
  • 23. Example 1 Using only the numbers from {-3, -2, -1, 0, 1, 2, 3}, determine the solution(s) of each equation. 2 a. 14x + 3 = 12x + 7 b. x = 9 c. 3x + 8 = 5 −12x −12x 2x + 3 = 7 −3 −3 2x = 4 2 2 x=2 14(2) + 3 = 12(2) + 7
  • 24. Example 1 Using only the numbers from {-3, -2, -1, 0, 1, 2, 3}, determine the solution(s) of each equation. 2 a. 14x + 3 = 12x + 7 b. x = 9 c. 3x + 8 = 5 −12x −12x 2x + 3 = 7 −3 −3 2x = 4 2 2 x=2 14(2) + 3 = 12(2) + 7 28 + 3 = 24 + 7
  • 25. Example 1 Using only the numbers from {-3, -2, -1, 0, 1, 2, 3}, determine the solution(s) of each equation. 2 a. 14x + 3 = 12x + 7 b. x = 9 c. 3x + 8 = 5 −12x −12x 2x + 3 = 7 2 x =± 9 −3 −3 2x = 4 2 2 x=2 14(2) + 3 = 12(2) + 7 28 + 3 = 24 + 7
  • 26. Example 1 Using only the numbers from {-3, -2, -1, 0, 1, 2, 3}, determine the solution(s) of each equation. 2 a. 14x + 3 = 12x + 7 b. x = 9 c. 3x + 8 = 5 −12x −12x 2x + 3 = 7 2 x =± 9 −3 −3 2x = 4 x = ±3 2 2 x=2 14(2) + 3 = 12(2) + 7 28 + 3 = 24 + 7
  • 27. Example 1 Using only the numbers from {-3, -2, -1, 0, 1, 2, 3}, determine the solution(s) of each equation. 2 a. 14x + 3 = 12x + 7 b. x = 9 c. 3x + 8 = 5 −12x −12x 2x + 3 = 7 2 x =± 9 −3 −3 2x = 4 x = ±3 2 2 2 x=2 3 =9 14(2) + 3 = 12(2) + 7 28 + 3 = 24 + 7
  • 28. Example 1 Using only the numbers from {-3, -2, -1, 0, 1, 2, 3}, determine the solution(s) of each equation. 2 a. 14x + 3 = 12x + 7 b. x = 9 c. 3x + 8 = 5 −12x −12x 2x + 3 = 7 2 x =± 9 −3 −3 2x = 4 x = ±3 2 2 2 x=2 3 =9 2 (−3) = 9 14(2) + 3 = 12(2) + 7 28 + 3 = 24 + 7
  • 29. Example 1 Using only the numbers from {-3, -2, -1, 0, 1, 2, 3}, determine the solution(s) of each equation. 2 a. 14x + 3 = 12x + 7 b. x = 9 c. 3x + 8 = 5 −12x −12x −8 −8 2x + 3 = 7 2 x =± 9 −3 −3 2x = 4 x = ±3 2 2 2 x=2 3 =9 2 (−3) = 9 14(2) + 3 = 12(2) + 7 28 + 3 = 24 + 7
  • 30. Example 1 Using only the numbers from {-3, -2, -1, 0, 1, 2, 3}, determine the solution(s) of each equation. 2 a. 14x + 3 = 12x + 7 b. x = 9 c. 3x + 8 = 5 −12x −12x −8 −8 2x + 3 = 7 2 x =± 9 3x = −3 −3 −3 2x = 4 x = ±3 2 2 2 x=2 3 =9 2 (−3) = 9 14(2) + 3 = 12(2) + 7 28 + 3 = 24 + 7
  • 31. Example 1 Using only the numbers from {-3, -2, -1, 0, 1, 2, 3}, determine the solution(s) of each equation. 2 a. 14x + 3 = 12x + 7 b. x = 9 c. 3x + 8 = 5 −12x −12x −8 −8 2x + 3 = 7 2 x =± 9 3x = −3 −3 −3 3 3 2x = 4 x = ±3 2 2 2 x=2 3 =9 2 (−3) = 9 14(2) + 3 = 12(2) + 7 28 + 3 = 24 + 7
  • 32. Example 1 Using only the numbers from {-3, -2, -1, 0, 1, 2, 3}, determine the solution(s) of each equation. 2 a. 14x + 3 = 12x + 7 b. x = 9 c. 3x + 8 = 5 −12x −12x −8 −8 2x + 3 = 7 2 x =± 9 3x = −3 −3 −3 3 3 2x = 4 x = ±3 x = −1 2 2 2 x=2 3 =9 2 (−3) = 9 14(2) + 3 = 12(2) + 7 28 + 3 = 24 + 7
  • 33. Example 1 Using only the numbers from {-3, -2, -1, 0, 1, 2, 3}, determine the solution(s) of each equation. 2 a. 14x + 3 = 12x + 7 b. x = 9 c. 3x + 8 = 5 −12x −12x −8 −8 2x + 3 = 7 2 x =± 9 3x = −3 −3 −3 3 3 2x = 4 x = ±3 x = −1 2 2 2 x=2 3 =9 3(−1) + 8 = 5 2 (−3) = 9 14(2) + 3 = 12(2) + 7 28 + 3 = 24 + 7
  • 34. Example 1 Using only the numbers from {-3, -2, -1, 0, 1, 2, 3}, determine the solution(s) of each equation. 2 a. 14x + 3 = 12x + 7 b. x = 9 c. 3x + 8 = 5 −12x −12x −8 −8 2x + 3 = 7 2 x =± 9 3x = −3 −3 −3 3 3 2x = 4 x = ±3 x = −1 2 2 2 x=2 3 =9 3(−1) + 8 = 5 2 (−3) = 9 −3 + 8 = 5 14(2) + 3 = 12(2) + 7 28 + 3 = 24 + 7
  • 35. Example 2 Solve each equation. a. b − 3 = 14 b. 45m = 9 c. 4w + 7 = 23
  • 36. Example 2 Solve each equation. a. b − 3 = 14 b. 45m = 9 c. 4w + 7 = 23 +3 +3
  • 37. Example 2 Solve each equation. a. b − 3 = 14 b. 45m = 9 c. 4w + 7 = 23 +3 +3 b = 17
  • 38. Example 2 Solve each equation. a. b − 3 = 14 b. 45m = 9 c. 4w + 7 = 23 +3 +3 b = 17 17 − 3 = 14
  • 39. Example 2 Solve each equation. a. b − 3 = 14 b. 45m = 9 c. 4w + 7 = 23 +3 +3 45 45 b = 17 17 − 3 = 14
  • 40. Example 2 Solve each equation. a. b − 3 = 14 b. 45m = 9 c. 4w + 7 = 23 +3 +3 45 45 b = 17 1 m= 5 17 − 3 = 14
  • 41. Example 2 Solve each equation. a. b − 3 = 14 b. 45m = 9 c. 4w + 7 = 23 +3 +3 45 45 b = 17 1 m= 5 17 − 3 = 14 1 (45) = 9 5
  • 42. Example 2 Solve each equation. a. b − 3 = 14 b. 45m = 9 c. 4w + 7 = 23 +3 +3 45 45 b = 17 1 m= 5 17 − 3 = 14 1 (45) = 9 5 45 =9 5
  • 43. Example 2 Solve each equation. a. b − 3 = 14 b. 45m = 9 c. 4w + 7 = 23 +3 +3 45 45 −7 −7 b = 17 1 m= 5 17 − 3 = 14 1 (45) = 9 5 45 =9 5
  • 44. Example 2 Solve each equation. a. b − 3 = 14 b. 45m = 9 c. 4w + 7 = 23 +3 +3 45 45 −7 −7 b = 17 1 4w = 16 m= 5 17 − 3 = 14 1 (45) = 9 5 45 =9 5
  • 45. Example 2 Solve each equation. a. b − 3 = 14 b. 45m = 9 c. 4w + 7 = 23 +3 +3 45 45 −7 −7 b = 17 1 4w = 16 m= 4 4 5 17 − 3 = 14 1 (45) = 9 5 45 =9 5
  • 46. Example 2 Solve each equation. a. b − 3 = 14 b. 45m = 9 c. 4w + 7 = 23 +3 +3 45 45 −7 −7 b = 17 1 4w = 16 m= 4 4 5 17 − 3 = 14 w=4 1 (45) = 9 5 45 =9 5
  • 47. Example 2 Solve each equation. a. b − 3 = 14 b. 45m = 9 c. 4w + 7 = 23 +3 +3 45 45 −7 −7 b = 17 1 4w = 16 m= 4 4 5 17 − 3 = 14 w=4 1 (45) = 9 5 4(4) + 7 = 23 45 =9 5
  • 48. Example 2 Solve each equation. a. b − 3 = 14 b. 45m = 9 c. 4w + 7 = 23 +3 +3 45 45 −7 −7 b = 17 1 4w = 16 m= 4 4 5 17 − 3 = 14 w=4 1 (45) = 9 5 4(4) + 7 = 23 45 16 + 7 = 23 =9 5
  • 49. Example 3 State what the first step to solving each equation would be and how you know you should do that step. Then solve. a. 4x + 5 = 3x + 6 b. 18a − 7 = 101
  • 50. Example 3 State what the first step to solving each equation would be and how you know you should do that step. Then solve. a. 4x + 5 = 3x + 6 b. 18a − 7 = 101 Subtract 3x from both sides to get all of the x’s on the same side of the equation
  • 51. Example 3 State what the first step to solving each equation would be and how you know you should do that step. Then solve. a. 4x + 5 = 3x + 6 b. 18a − 7 = 101 Subtract 3x from both sides to get all of the x’s on the same side of the equation 4x + 5 = 3x + 6
  • 52. Example 3 State what the first step to solving each equation would be and how you know you should do that step. Then solve. a. 4x + 5 = 3x + 6 b. 18a − 7 = 101 Subtract 3x from both sides to get all of the x’s on the same side of the equation 4x + 5 = 3x + 6 −3x −3x
  • 53. Example 3 State what the first step to solving each equation would be and how you know you should do that step. Then solve. a. 4x + 5 = 3x + 6 b. 18a − 7 = 101 Subtract 3x from both sides to get all of the x’s on the same side of the equation 4x + 5 = 3x + 6 −3x − 5 −3x − 5
  • 54. Example 3 State what the first step to solving each equation would be and how you know you should do that step. Then solve. a. 4x + 5 = 3x + 6 b. 18a − 7 = 101 Subtract 3x from both sides to get all of the x’s on the same side of the equation 4x + 5 = 3x + 6 −3x − 5 −3x − 5
  • 55. Example 3 State what the first step to solving each equation would be and how you know you should do that step. Then solve. a. 4x + 5 = 3x + 6 b. 18a − 7 = 101 Subtract 3x from both sides to get all of the x’s on the same side of the equation 4x + 5 = 3x + 6 −3x − 5 −3x − 5 x=1
  • 56. Example 3 State what the first step to solving each equation would be and how you know you should do that step. Then solve. a. 4x + 5 = 3x + 6 b. 18a − 7 = 101 Subtract 3x from both sides to get all of the x’s on the same side of the equation 4x + 5 = 3x + 6 −3x − 5 −3x − 5 x=1 Check!
  • 57. Example 3 State what the first step to solving each equation would be and how you know you should do that step. Then solve. a. 4x + 5 = 3x + 6 b. 18a − 7 = 101 Subtract 3x from both Add 7 to both sids to sides to get all of the get the variable term x’s on the same side of by itself the equation 4x + 5 = 3x + 6 −3x − 5 −3x − 5 x=1 Check!
  • 58. Example 3 State what the first step to solving each equation would be and how you know you should do that step. Then solve. a. 4x + 5 = 3x + 6 b. 18a − 7 = 101 Subtract 3x from both Add 7 to both sids to sides to get all of the get the variable term x’s on the same side of by itself the equation 18a − 7 = 101 4x + 5 = 3x + 6 −3x − 5 −3x − 5 x=1 Check!
  • 59. Example 3 State what the first step to solving each equation would be and how you know you should do that step. Then solve. a. 4x + 5 = 3x + 6 b. 18a − 7 = 101 Subtract 3x from both Add 7 to both sids to sides to get all of the get the variable term x’s on the same side of by itself the equation 18a − 7 = 101 4x + 5 = 3x + 6 +7 +7 −3x − 5 −3x − 5 x=1 Check!
  • 60. Example 3 State what the first step to solving each equation would be and how you know you should do that step. Then solve. a. 4x + 5 = 3x + 6 b. 18a − 7 = 101 Subtract 3x from both Add 7 to both sids to sides to get all of the get the variable term x’s on the same side of by itself the equation 18a − 7 = 101 4x + 5 = 3x + 6 +7 +7 −3x − 5 −3x − 5 18a = 108 x=1 Check!
  • 61. Example 3 State what the first step to solving each equation would be and how you know you should do that step. Then solve. a. 4x + 5 = 3x + 6 b. 18a − 7 = 101 Subtract 3x from both Add 7 to both sids to sides to get all of the get the variable term x’s on the same side of by itself the equation 18a − 7 = 101 4x + 5 = 3x + 6 +7 +7 −3x − 5 −3x − 5 18a = 108 x=1 18 18 Check!
  • 62. Example 3 State what the first step to solving each equation would be and how you know you should do that step. Then solve. a. 4x + 5 = 3x + 6 b. 18a − 7 = 101 Subtract 3x from both Add 7 to both sids to sides to get all of the get the variable term x’s on the same side of by itself the equation 18a − 7 = 101 4x + 5 = 3x + 6 +7 +7 −3x − 5 −3x − 5 18a = 108 x=1 18 18 Check! a=6
  • 63. Example 3 State what the first step to solving each equation would be and how you know you should do that step. Then solve. a. 4x + 5 = 3x + 6 b. 18a − 7 = 101 Subtract 3x from both Add 7 to both sids to sides to get all of the get the variable term x’s on the same side of by itself the equation 18a − 7 = 101 4x + 5 = 3x + 6 +7 +7 −3x − 5 −3x − 5 18a = 108 x=1 18 18 Check! a=6 Check!
  • 64. Example 3 State what the first step to solving each equation would be and how you know you should do that step. Then solve. c. 6(x − 5) = 42 d. 7g − 12 = −40
  • 65. Example 3 State what the first step to solving each equation would be and how you know you should do that step. Then solve. c. 6(x − 5) = 42 d. 7g − 12 = −40 Divide by 6 to work on getting the variable isolated
  • 66. Example 3 State what the first step to solving each equation would be and how you know you should do that step. Then solve. c. 6(x − 5) = 42 d. 7g − 12 = −40 Divide by 6 to work on getting the variable isolated 6(x − 5) = 42
  • 67. Example 3 State what the first step to solving each equation would be and how you know you should do that step. Then solve. c. 6(x − 5) = 42 d. 7g − 12 = −40 Divide by 6 to work on getting the variable isolated 6(x − 5) = 42 6 6
  • 68. Example 3 State what the first step to solving each equation would be and how you know you should do that step. Then solve. c. 6(x − 5) = 42 d. 7g − 12 = −40 Divide by 6 to work on getting the variable isolated 6(x − 5) = 42 6 6 x−5=7
  • 69. Example 3 State what the first step to solving each equation would be and how you know you should do that step. Then solve. c. 6(x − 5) = 42 d. 7g − 12 = −40 Divide by 6 to work on getting the variable isolated 6(x − 5) = 42 6 6 x−5=7 +5 +5
  • 70. Example 3 State what the first step to solving each equation would be and how you know you should do that step. Then solve. c. 6(x − 5) = 42 d. 7g − 12 = −40 Divide by 6 to work on getting the variable isolated 6(x − 5) = 42 6 6 x−5=7 +5 +5 x = 12
  • 71. Example 3 State what the first step to solving each equation would be and how you know you should do that step. Then solve. c. 6(x − 5) = 42 d. 7g − 12 = −40 Divide by 6 to work Add 12 to isolate the on getting the variable term variable isolated 6(x − 5) = 42 6 6 x−5=7 +5 +5 x = 12
  • 72. Example 3 State what the first step to solving each equation would be and how you know you should do that step. Then solve. c. 6(x − 5) = 42 d. 7g − 12 = −40 Divide by 6 to work Add 12 to isolate the on getting the variable term variable isolated 7g − 12 = −40 6(x − 5) = 42 6 6 x−5=7 +5 +5 x = 12
  • 73. Example 3 State what the first step to solving each equation would be and how you know you should do that step. Then solve. c. 6(x − 5) = 42 d. 7g − 12 = −40 Divide by 6 to work Add 12 to isolate the on getting the variable term variable isolated 7g − 12 = −40 6(x − 5) = 42 +12 +12 6 6 x−5=7 +5 +5 x = 12
  • 74. Example 3 State what the first step to solving each equation would be and how you know you should do that step. Then solve. c. 6(x − 5) = 42 d. 7g − 12 = −40 Divide by 6 to work Add 12 to isolate the on getting the variable term variable isolated 7g − 12 = −40 6(x − 5) = 42 +12 +12 6 6 7g = −28 x−5=7 +5 +5 x = 12
  • 75. Example 3 State what the first step to solving each equation would be and how you know you should do that step. Then solve. c. 6(x − 5) = 42 d. 7g − 12 = −40 Divide by 6 to work Add 12 to isolate the on getting the variable term variable isolated 7g − 12 = −40 6(x − 5) = 42 +12 +12 6 6 7g = −28 x−5=7 7 7 +5 +5 x = 12
  • 76. Example 3 State what the first step to solving each equation would be and how you know you should do that step. Then solve. c. 6(x − 5) = 42 d. 7g − 12 = −40 Divide by 6 to work Add 12 to isolate the on getting the variable term variable isolated 7g − 12 = −40 6(x − 5) = 42 +12 +12 6 6 7g = −28 x−5=7 7 7 +5 +5 x = 12 g = −4
  • 77. Example 4 The formula for converting from degrees Celsius to degrees Fahrenheit is given below. Convert 43°C to °F. 9 F = C + 32 5
  • 78. Example 4 The formula for converting from degrees Celsius to degrees Fahrenheit is given below. Convert 43°C to °F. 9 F = C + 32 5 9 F = (43) + 32 5
  • 79. Example 4 The formula for converting from degrees Celsius to degrees Fahrenheit is given below. Convert 43°C to °F. 9 F = C + 32 5 9 F = (43) + 32 5 F = 77.4 + 32
  • 80. Example 4 The formula for converting from degrees Celsius to degrees Fahrenheit is given below. Convert 43°C to °F. 9 F = C + 32 5 9 F = (43) + 32 5 F = 77.4 + 32 F = 109.4
  • 81. Example 4 The formula for converting from degrees Celsius to degrees Fahrenheit is given below. Convert 43°C to °F. 9 F = C + 32 5 9 F = (43) + 32 5 F = 77.4 + 32 F = 109.4 43°C is 109.4°F
  • 82. Problem Set
  • 83. Problem Set p. 106 #1-37 odd “The greatest challenge to any thinker is stating the problem in a way that will allow a solution.” - Bertrand Russell