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3. Translating Equations

4. Objectives Students will translate and write equations into mathematical form. Students will translate and write equations from mathematical forms.

5. Equation Verbal expressions that involve the equal signs. Verbal expressions are mathematical statements that are written out into words.

6. Equals Sign Is Equals Is equal to Is the same as Is as much as Is identical to

7. Example 1-1a Translate this sentence into an equation. A number b divided by three is equal to six less than c. b divided by three is equal to six less than c. Answer: The equation is.

8. Example 1-1b Translate this sentence into an equation. Fifteen more than z times six is y times two minus eleven. 15z6y211

9. Answer: The equation is. Example 1-1c Translate each sentence into an equation. a.A number c multiplied by six is equal to two more than d. b.Three less than a number a divided by four is seven more than 3 times b.

10. Answer: The formula is. Example 1-3a Translate the sentence into a formula. WordsPerimeter equals four times the length of the side. VariablesLet P = perimeter and s = length of a side. The perimeter of a square equals four times the length of the side. Perimeter equalsfour times the length of a side. P 4s

11. Answer: The formula is. Example 1-3b Translate the sentence into a formula. The area of a circle equals the product of  and the square of the radius r.

12. Example 1-4a Translate this equation into a verbal sentence. Answer: Twelve minus two times x equals negative five. Twelve minus two times x equals negative five. 12 2x 5

13. Example 1-4b Translate this equation into a verbal sentence. Answer: a squared plus three times b equals c divided by six. a squared plus three times b equals c divided by six. a 2 3b

14. Example 1-4c Translate each equation into a verbal sentence. Answer:Twelve divided by b minus four equals negative one. Answer:Five times a equals b squared plus one. 1. 2.

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17. Adding and Subtracting Equations

18. Objectives Solve equations by adding and subtracting. Write and solve an equation for given conditions. Write an equation to solve a word problem.

19. To Solve an equation… 1)Start on the same side of your variable. 2)Move your number from left to right. a)Opposite operation. i.Addition → Subtraction ii.Subtraction → Addition

20. Solve. Then check your solution. Example 2-1a Original equation Answer: and To check that –15 is the solution, substitute –15 for h in the original equation. Add 12 to each side.

21. Solve. Then check your solution. Example 2-1b Answer: 40 Check:

22. Solve. Then check your solution. Example 2-2a To check that 29 is the solution, substitute 29 for k in the original equation. Start on the same side as the variable. Answer: Subtract 63 to each side.

23. Solve. Then check your solution. Example 2-2b Answer: –61 Check:

24. Solve. Then check your solution. Example 2-3a To check that –66 is the solution, substitute –66 for c in the original equation. Original equation Answer: and Subtract 102 from each side.

25. Solve. Then check your solution. Example 2-3b Answer: –171 Check:

26. Example 2-4a Solvein two ways. Method 1 Use the Subtraction Property of Equality. Original equation and Answer: or Subtractfrom each side.

27. Example 2-4c Solve. Answer:

28. Example 2-5a Write an equation for the problem. Then solve the equation and check your solution. Fourteen more than a number is equal to twenty-seven. Find this number. Fourteen more than a number is equal to twenty-seven. Original equation 14 n 27 Subtract 14 from each side.

29. Example 2-5b To check that 13 is the solution, substitute 13 for n in the original equation. The solution is 13. and Answer:

30. Example 2-5c Twelve less than a number is equal to negative twenty-five. Find the number. Answer: –13

31. Example 2-6a History The Washington Monument in Washington, D.C., was built in two phases. From 1848–1854, the monument was built to a height of 152 feet. From 1854 until 1878, no work was done. Then from 1878 to 1888, the additional construction resulted in its final height of 555 feet. How much of the monument was added during the second construction phase?

32. Answer:There were 403 feet added to the Washington Monument from 1878 to 1888. Example 2-6b WordsThe first height plus the additional height equals 555 feet. VariablesLet a = the additional height. The first height plus the additional height equals 555. Original equation and 152 a 555 Subtract 152 from each side.

33. Example 2-6c The Sears Tower was built in 1974. The height to the Sky Deck is 1353 feet. The actual recorded height is 1450 feet. In 1982, they added twin antenna towers, which does not count for the record, for a total structure height of 1707 feet. How tall are the twin antenna towers? Answer: 257 feet

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36. Multiplying and Dividing Equations

37. Objectives Solve equations using multiplication and division. Write and solve equations from expressions and word problem using multiplication and division.

38.  Start on the same as the variable.  Divide Coefficient to Both Sides  IF you have a division bar with a VARIABLE ON TOP and NO ADDITION/SUBTRACTION SIGN, then turn the NUMBERS into a fraction multiplied by a variable.

39. Solve. Then check your solution. Example 3-1a Answer: Divide your coefficient to both sides.

40. Solve. Example 3-1b Answer: 12

41. Solve. Then check your solution. Example 3-2a Answer: or Check this result. Start on the same side as your variable. Divide your coefficient to both sides.

42. Example 3-2c Solve. Answer:

43. Solve. Example 3-3a To check that 5 is the solution, substitute 5 for b in the original equation. Start on the same side as your variable. Answer: Check this result. Divide each side by -15.

44. Solve. Example 3-3b Answer:

45. Solve. Then check your solution. Example 3-5a Original equation Answer: and To check, substitute 13 for w. Divide each side by 11.

46. Solve. Then check your solution. Check: Answer: 17 Example 3-5b

47. Solve. Example 3-6a Original equation and Answer: Divide each side by –8.

48. Solve. Example 3-6b Answer: –23

49. Example 3-7a Write an equation for the problem below. Then solve the equation. Negative fourteen times a number equals 224. Original equation Answer: Check this result. Negative fourteen times a number equals 224. n–14224 Divide each side by –14.

50. Negative thirty-four times a number equals 578. Find the number. Example 3-7b Answer: –17

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53. Two-Step Equations

54. Objectives Solve word problems by working backwards. Solve two-step equations using multiplication, addition, subtraction, and division. Write and solve two-step equations.

55. Example 4-1a Danny took some rope with him on his camping trip. He used 32 feet of rope to tie his canoe to a log on the shore. The next night, he used half of the remaining rope to secure his tent during a thunderstorm. On the last day, he used 7 feet as a fish stringer to keep the fish that he caught. After the camping trip, he had 9 feet left. How much rope did he have at the beginning of the camping trip?

56. Example 4-1b Start at the end of the problem and undo each step. StatementUndo the Statement He had 9 feet left. He used 7 feet as a fish stringer. He used half of the remaining rope to secure his tent. He used 32 feet to tie his canoe. 9 9 + 7 = 16 16  2 = 32 32 + 32 = 64 Answer:He had 64 feet of rope. Check the answer in the context of the problem.

57. Example 4-1c Olivia went to the mall to spend some of her monthly allowance. She put $10 away so it could be deposited in the savings account at a later date. The first thing she bought was a CD for $15.99. The next stop was to buy hand lotion and a candle, which set her back $9.59. For lunch, she spent half of the remaining cash. She went to the arcade room and spent $5.00 and took home $1.21. How much was Olivia’s monthly allowance? Answer: $48.00

58. Two-Step Equations Start on the same side as your variable. Move numbers from left to right. Opposite Operation –Addition ↔ Subtraction Divide the coefficient to both sides. –IF you have a decimal bar, change the numbers to a fraction. Then divide.

59. Solve. Then check your solution. Example 4-2a Simplify. Answer: Simplify. To check, substitute 10 for q in the original equation. Original equation Add 13 to each side. Divide each side by 5.

60. Solve. Example 4-2b Answer: 14

61. Solve. Then check your solution. Example 4-3a Original equation Answer: s = –240 Simplify. Subtract 9 from each side. =

62. Solve. Example 4-3c Answer: 363

63. Solve. Example 4-4a Since there is a fraction bar with a subtraction sign on top, split and divide. Answer: -2

64. Example 4-4b Solve. Answer: 21

65. Example 4-5a Write an equation for the problem below. Then solve the equation. Eight more than five times a number is negative 62. Original equation Simplify. Eight more than five times a number is negative 62. n8 62 5 Subtract 8 from each side.

66. Example 4-5b Simplify. Answer: n = –14 Divide each side by 5.

67. Example 4-5c Three-fourths of seven subtracted from a number is negative fifteen. What is the number? Answer: –13

68. Example 4-6a Number Theory Write an equation for the problem below. Then solve the equation and answer the problem. Find three consecutive odd integers whose sum is 57. Let n = the least odd integer. Let n + 2 = the next greater odd integer. Let n + 4 = the greatest of the three odd integers. The sum of three consecutive odd integers is 57. =57

69. Example 4-6b Simplify. Original equation Simplify. or 19 or 21 Answer: The consecutive odd integers are 17, 19, and 21. Subtract 6 from each side. Divide each side by 3.

70. Example 4-6c Find three consecutive even integers whose sum is 84. Answer: 26, 28, 30

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73. Multi-Step Equations

74. Objectives Solve multi-step equations. Solve equations with grouping symbols. Define no solution and infinitely many solution. Identify equations with no solutions and infinitely many solutions.

75. Multi-Step Equations Parentheses/Distributive Property/Fraction Bar Combine Same-Side Like Terms –Same operation that you are given. Combine Opposite-Side Like Terms Opposite Operation –Move numbers from left to right. –Move variables from right to left Divide Coefficient To Both Sides

76. Other Solutions No Solution No variable exists Values are unequal Also seen as: –{ } –{ø} –ø Infinitely Many Solutions No variable exists Values are equal Also known as: –All reals –All real numbers –All numbers

77. Solve. Example 5-1a Original equation Simplify. Answer: s = 5 Simplify. Subtract 7s from each side. Subtract 8 from each side. Divide each side by –2.

78. Solve. Example 5-1c Answer:

79. Solving Equations—Calculator 1.MENU 2.8: Equations 3.F3 4.Type Equation in 1.Use “X” for ALL variables. 2.Press SHIFT. for = 5.Enter 6.Enter IF it comes up ERROR, then you: 1.Need to type it in again and press ENTER, ENTER. 2.Solve by hand to see if it is NO SOLUTION or ALL REALS (infinitely many)

80. Solve. Then check your solution. Example 5-2a Original equation Distributive Property Simplify. Subtract 12q from each side. Subtract 6 from each side.

81. Example 5-2b Simplify. Answer: q = 6 Simplify. To check, substitute 6 for q in the original equation. Divide each side by –8.

82. Solve. Example 5-2c Answer: 36

83. Solve. Example 5-3a Original equation Distributive Property This statement is false. Answer:There must be at least one c to represent the variable. This equation has no solution. Subtract 40c from each side.

84. Example 5-3b Solve. Answer: This equation has no solution.

85. Solve. Then check your solution. Example 5-4a Original equation Distributive Property Answer: Since the expression on each side of the equation is the same, this equation is an identity. The statementis true for all values of t.

86. Answer: is true for all values of c. Example 5-4b Solve.

87. Multiple-Choice Test Item Solve. A 13 B –13 C 26 D –26 Example 5-5a Read the Test Item You are asked to solve an equation. Solve the Test Item You can solve the equation or substitute each value into the equation and see if it makes the equation true. We will solve by substitution.

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90. Ratios and Proportions

91. Objectives Define ratio and proportion. Identify proportion and determine whether ratios form a proportion. Use cross-products to solve proportions for a given variable.

92. Example 6-1a Answer:The ratios are equal. Therefore, they form a proportion. Determine whether the ratios and form a proportion. Type each ratio into the calculator, and press enter. If they equal, then they do not form a proportion.

93. Do the ratios and form a proportion? Answer:The ratios are not equal. Therefore, they do not form a proportion. Example 6-1b Type each ratio into the calculator, and press enter. If they equal, then they do not form a proportion.

94. Example 6-3a Solve the proportion. Original equation Find the cross products. Simplify. Divide each side by 8. Answer: Simplify.

95. Example 6-3b Solve the proportion. Answer: 6.3

96. Example 6-4a ExploreLet p represent the number of times needed to crank the pedals. PlanWrite a proportion for the problem. turns of the pedals wheel turns turns of the pedals wheel turns Bicycling The gear on a bicycle is 8:5. This means that for every eight turns of the pedals, the wheel turns five times. Suppose the bicycle wheel turns about 2435 times during a trip. How many times would you have to turn the pedals during the trip?

97. Solve Original proportion Example 6-4b Find the cross products. Divide each side by 5. Answer: 3896 = p Simplify.

98. Example 6-4c ExamineIf it takes 8 turns of the pedal to make the wheel turn 5 times, then it would take 1.6 turns of the pedal to make the wheel turn 1 time. So, if the wheel turns 2435 times, then there are 2435  1.6 or 3896 turns of the pedal. The answer is correct.

99. Example 6-4d Before 1980, Disney created animated movies using cels. These hand drawn cels (pictures) of the characters and scenery represented the action taking place, one step at a time. For the movie Snow White, it took 24 cels per second to have the characters move smoothly. The movie is around 42 minutes long. About how many cels were drawn to produce Snow White? Answer:About 60,480 cels were drawn to produce Snow White.

100. Example 6-5a Maps In a road atlas, the scale for the map of Connecticut is 5 inches = 41 miles. The scale for the map of Texas is 5 inches = 144 miles. What are the distances in miles represented by 2.5 inches on each map? ExploreLet d represent the actual distance. PlanWrite a proportion for the problem. scale actual scale actual Connecticut:

101. Example 6-5b Texas: scale actual scale actual

102. Example 6-5c Solve Connecticut: Find the cross products. Simplify. Divide each side by 5. Simplify. or 20.5

103. Example 6-5d Solve Texas: Find the cross products. Simplify. Divide each side by 5. Simplify. or 72

104. Example 6-5f Answer:The actual distance in Connecticut represented by 2.5 inches is 20.5 miles. The actual distance in Texas represented by 2.5 inches is 72 miles. Examine: 2.5 inches isof 5 inches. So 2.5 inches represents (41) or 20.5 miles in Connecticut and (144) or 72 miles in Texas. The answer is correct.

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107. Percent of Change

108. Objective Use to the percent of change formula to determine the amount of change an item has. Determine whether an item is a percent of increase or a percent of decrease. Find the amount an item is by taking the tax and/or discount

109. Example 7-1a State whether the percent of change is a percent of increase or a percent of decrease. Then find the percent of change. original: 32 new: 40

110. Example 7-1c State whether the percent of change is a percent of increase or a percent of decrease. Then find the percent of change. original: 20 new: 4

111. Example 7-1e a.State whether the percent of change is a percent of increase or a percent of decrease. Then find the percent of change. original: 20 new: 18

112. Example 7-1f b.State whether the percent of change is a percent of increase or a percent of decrease. Then find the percent of change. original: 12 new: 48

113. Example 7-3a Sales Tax A meal for two at a restaurant costs $32.75. If the sales tax is 5%, what is the total price of the meal? The tax is 5% of the price of the meal. Use a calculator. Round $1.6375 to $1.64. Add this amount to the original price. Answer: The total price of the meal is $34.49.

114. Example 7-3b A portable CD player costs $69.99. If the sales tax is 6.75%, what is the total price of the CD player? Answer: The total price of the CD player is $74.71.

115. Example 7-4a The discount is 20% of the original price. Subtract $0.76 from the original price. Answer:The discounted price of the dog toy is $3.04. Discount A dog toy is on sale for 20% off the original price. If the original price of the toy is $3.80, what is the discounted price?

116. Example 7-4b A baseball cap is on sale for 15% off the original price. If the original price of the cap is $19.99, what is the discounted price? Answer: The discounted price of the cap player is $16.99.

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