Equations and Formulas

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. Homework

83. Homework
p. 106 #1-9 all, 11-37 odd
“The greatest challenge to any thinker is stating the problem in a way that
will allow a solution.” - Bertrand Russell