1. Adapted from Pearson Education, Inc. Copyright © 2009 Pearson Education, Inc. Welcome to MM150! Kirsten K. Meymaris Unit 3 Plan for the hour Order of Operations (3.1) Learning the Rules Linear Equations, One Variable (3.2) Setting up the Game Formulas (3.3) Playing the Game Applications of Linear Equations, One Variable (3.4) New Games

2. Adapted from Pearson Education, Inc. Copyright © 2009 Pearson Education, Inc. 3.1 Order of Operations

3. Adapted from Pearson Education, Inc. Copyright © 2009 Pearson Education, Inc. Order of Operations 1.All operations within Parentheses or other grouping symbols (according to the following order). 2.All Exponential operations (raising to powers or finding roots). 3.All Multiplication and Divisions - left to right. 4.Additions and Subtractions - left to right. PEMDAS

4. Adapted from Pearson Education, Inc. Copyright © 2009 Pearson Education, Inc. Order of Operations 1.All operations within Parentheses or other grouping symbols (according to the following order). 2.All Exponential operations (raising to powers or finding roots). 3.All Multiplication and Divisions - left to right. 4.Additions and Subtractions - left to right. PEMDAS, Also known as: “Please Excuse My Dear Aunt Sally” “Place Everything Math Down Another Subway”

5. Adapted from Pearson Education, Inc. Copyright © 2009 Pearson Education, Inc. Example: Evaluating an Expression Evaluate the expression x 2 + 4x + 5 for x = 3.

6. Adapted from Pearson Education, Inc. Copyright © 2009 Pearson Education, Inc. Example: Evaluating an Expression Evaluate the expression x 2 + 4x + 5 for x = 3. Solution: x 2 + 4x + 5 = 3 2 + 4(3) + 5 = 9 + 12 + 5 = 26

7. Adapted from Pearson Education, Inc. Copyright © 2009 Pearson Education, Inc. Example: Evaluating an Expression Evaluate the expression: 7x 2 + 5x – 11 for x = -1

8. Adapted from Pearson Education, Inc. Copyright © 2009 Pearson Education, Inc. Example: Evaluating an Expression Evaluate the expression: 7x 2 + 5x – 11 for x = -1 Solution: = (7)(-1) 2 + (5)(-1) – 11 = (7)(1) + (5)(-1) – 11 = 7 + (-5) – 11 = 7 – 5 – 11 = -9

9. Adapted from Pearson Education, Inc. Copyright © 2009 Pearson Education, Inc. 3.2 Linear Equations in One Variable

10. Adapted from Pearson Education, Inc. Copyright © 2009 Pearson Education, Inc. Definitions Terms are parts that are added or subtracted in an algebraic expression. Coefficient is the numerical part of a term. Like terms are terms that have the same variables with the same exponents on the variables. Unlike terms have different variables or different exponents on the variables.

11. Adapted from Pearson Education, Inc. Copyright © 2009 Pearson Education, Inc. Properties of the Real Numbers Associative property of multiplication (ab)c = a(bc) Associative property of addition (a + b) + c = a + (b + c) Commutative property of multiplication ab = ba Commutative property of addition a + b = b + a Distributive propertya(b + c) = ab + ac

12. Adapted from Pearson Education, Inc. Copyright © 2009 Pearson Education, Inc. Example: Combine Like Terms 8x + 4x 5y  6y x + 15  5x + 9 3x + 2 + 6y  4 + 7x

13. Adapted from Pearson Education, Inc. Copyright © 2009 Pearson Education, Inc. Example: Combine Like Terms 8x + 4x = (8 + 4)x = 12x 5y  6y = (5  6)y =  y x + 15  5x + 9 = (1  5)x + (15 + 9) =  4x + 24 3x + 2 + 6y  4 + 7x = (3 + 7)x + 6y + (2  4) = 10x + 6y  2

14. Adapted from Pearson Education, Inc. Copyright © 2009 Pearson Education, Inc. Properties of Equality Addition Property of Equality If a = b, then a + c = b + c for all real numbers a, b, and c. Subtraction Property of Equality If a = b, then a  c = b  c for all real numbers a, b, and c. Multiplication Property of Equality If a = b, then a c = b c for all real numbers a, b, and c, where c  0. Division Property of Equality If a = b, then for all real numbers a, b, and c, c  0.

15. Adapted from Pearson Education, Inc. Copyright © 2009 Pearson Education, Inc. General Procedure for Solving Linear Equations Eliminate fractions with LCD (or LCM) Remove parentheses when necessary Combine like terms Collect all variables on one side, all constants on other Solve

16. Adapted from Pearson Education, Inc. Copyright © 2009 Pearson Education, Inc. Example: Solving Equations Solve 3x  4 = 17. Eliminate fractions with LCD (or LCM) Remove parentheses when necessary Combine like terms Collect all variables on one side, all constants on other Solve

17. Adapted from Pearson Education, Inc. Copyright © 2009 Pearson Education, Inc. Example: Solving Equations Solve 3x  4 = 17.

18. Adapted from Pearson Education, Inc. Copyright © 2009 Pearson Education, Inc. Example: Solving Equations Solve 8x + 3 = 6x + 21.

19. Adapted from Pearson Education, Inc. Copyright © 2009 Pearson Education, Inc. Example: Solving Equations Solve 8x + 3 = 6x + 21.

20. Adapted from Pearson Education, Inc. Copyright © 2009 Pearson Education, Inc. Example: Solving Equations Solve False, the equation has no solution. The equation is inconsistent.

21. Adapted from Pearson Education, Inc. Copyright © 2009 Pearson Education, Inc. Example: Solving Equations Solve True, 0 = 0 the solution is all real numbers.

22. Adapted from Pearson Education, Inc. Copyright © 2009 Pearson Education, Inc. Proportions A proportion is a statement of equality between two ratios. Cross Multiplication If then ad = bc, b  0, d  0.

23. Adapted from Pearson Education, Inc. Copyright © 2009 Pearson Education, Inc. Example If the ratio of boys to girls is 1:3 and there are 213 girls present, how many boys are present?

24. Adapted from Pearson Education, Inc. Copyright © 2009 Pearson Education, Inc. Example If the ratio of boys to girls is 1:3 and there are 213 girls present, how many boys are present?

25. Adapted from Pearson Education, Inc. Copyright © 2009 Pearson Education, Inc. Example If my Subaru can go 315 miles on about 1 tank of gas (13 gallons of gas), how many gallons of gas are needed for my trip from Atlanta to New York City (850 miles)?

26. Adapted from Pearson Education, Inc. Copyright © 2009 Pearson Education, Inc. Example If my Subaru can go 315 miles on about 1 tank of gas (13 gallons of gas), how many gallons of gas are needed for my trip from Atlanta to New York City (850 miles)?

27. Adapted from Pearson Education, Inc. Copyright © 2009 Pearson Education, Inc. Example A 50 pound bag of fertilizer will cover an area of 15,000 ft 2. How many pounds are needed to cover an area of 226,000 ft 2 ?

28. Adapted from Pearson Education, Inc. Copyright © 2009 Pearson Education, Inc. Example A 50 pound bag of fertilizer will cover an area of 15,000 ft 2. How many pounds are needed to cover an area of 226,000 ft 2 ? 754 pounds of fertilizer would be needed.

29. Adapted from Pearson Education, Inc. Copyright © 2009 Pearson Education, Inc. Examples – Open Forum!

30. Adapted from Pearson Education, Inc. Copyright © 2009 Pearson Education, Inc. 3.3 Formulas

31. Adapted from Pearson Education, Inc. Copyright © 2009 Pearson Education, Inc. Exponential Equations: Carbon Dating Carbon dating is used by scientists to find the age of fossils, bones, and other items. The formula used in carbon dating is

32. Adapted from Pearson Education, Inc. Copyright © 2009 Pearson Education, Inc. If 15 mg of C 14 is present in an animal bone recently excavated, how many milligrams will be present in 4000 years? Example

33. Adapted from Pearson Education, Inc. Copyright © 2009 Pearson Education, Inc. If 15 mg of C 14 is present in an animal bone recently excavated, how many milligrams will be present in 4000 years? Example P 0 = original amount = P= amount after t number of years = t = number of years = P = P 0 2 -t/5600

34. Adapted from Pearson Education, Inc. Copyright © 2009 Pearson Education, Inc. If 15 mg of C 14 is present in an animal bone recently excavated, how many milligrams will be present in 4000 years? Example P 0 = original amount = 15 mg P= amount after t number of years = ? t = number of years = 4000 P = P 0 2 -t/5600

35. Adapted from Pearson Education, Inc. Copyright © 2009 Pearson Education, Inc. Exponential Equations: Carbon Dating continued P = 15(2) -4000/5600

36. Adapted from Pearson Education, Inc. Copyright © 2009 Pearson Education, Inc. Example: Carbon Dating In 4000 years, approximately 9.2 mg of the original 15 mg of C 14 will remain.

37. Adapted from Pearson Education, Inc. Copyright © 2009 Pearson Education, Inc. Solving for a Variable in a Formula or Equation Solve the equation 3x + 8y  9 = 0 for y.

38. Adapted from Pearson Education, Inc. Copyright © 2009 Pearson Education, Inc. Solving for a Variable in a Formula or Equation Solve the equation 3x + 8y  9 = 0 for y.

39. Adapted from Pearson Education, Inc. Copyright © 2009 Pearson Education, Inc. 3.4 Applications of Linear Equations in One Variable ( Word Problems!)

40. Adapted from Pearson Education, Inc. Copyright © 2009 Pearson Education, Inc. Translating Words to Expressions Twice a number A number decreased by 8 Four less than a number A number increased by 5 Ten more than a number Mathematical Expression Phrase

41. Adapted from Pearson Education, Inc. Copyright © 2009 Pearson Education, Inc. Translating Words to Expressions 2x2x Twice a number x – 8 A number decreased by 8 x – 4Four less than a number x + 5A number increased by 5 x + 10Ten more than a number Mathematical Expression Phrase

42. Adapted from Pearson Education, Inc. Copyright © 2009 Pearson Education, Inc. Translating Words to Expressions Five less than 7 times a number The difference between a number and 6 2 decreased by a number Four times a number Mathematical Expression Phrase

43. Adapted from Pearson Education, Inc. Copyright © 2009 Pearson Education, Inc. Translating Words to Expressions Five less than 7 times a number x – 6 The difference between a number and 6 2 – x 2 decreased by a number 4x4xFour times a number Mathematical Expression Phrase 7x – 5

44. Adapted from Pearson Education, Inc. Copyright © 2009 Pearson Education, Inc. Translating Words to Expressions Twice a number, decreased by 3 is 8. Three less than a number is 4 Seven more than a number is 12 Mathematical Equation Phrase A number decreased by 15 is 9 times the number

45. Adapted from Pearson Education, Inc. Copyright © 2009 Pearson Education, Inc. Translating Words to Expressions 2x  3 = 8 Twice a number, decreased by 3 is 8. x – 3 = 4 Three less than a number is 4 x + 7 = 12 Seven more than a number is 12 Mathematical Equation Phrase x  15 = 9x A number decreased by 15 is 9 times the number

46. Adapted from Pearson Education, Inc. Copyright © 2009 Pearson Education, Inc. To Solve a Word Problem Read problem three (yes, 3!) times Sketch it Write what you know and want to know assign variables Write out equation Solve for unknown Answer the question(s) Include units Double check your answer

47. Adapted from Pearson Education, Inc. Copyright © 2009 Pearson Education, Inc. Example The bill (parts and labor) for the repairs of a car was $496.50. The cost of the parts was $339. The cost of the labor was $45 per hour. How many hours were billed?

48. Adapted from Pearson Education, Inc. Copyright © 2009 Pearson Education, Inc. Example The bill (parts and labor) for the repairs of a car was $496.50. The cost of the parts was $339. The cost of the labor was $45 per hour. How many hours were billed? h = # of hours billed parts + labor = total 339 + 45h = 496.50

49. Adapted from Pearson Education, Inc. Copyright © 2009 Pearson Education, Inc. Example continued The car was worked on for 3.5 hours.

50. Adapted from Pearson Education, Inc. Copyright © 2009 Pearson Education, Inc. The daycare my son attends charges $20.00 per day, 5 days a week. Children must be picked up by 6:00 pm otherwise they charge $5.00 for every 5 minutes you are late. I was 5 minutes late 3 times last week. How much did I pay that week? (Assuming my son went every day this week.) X = number of days at daycare = Y = number days late = T = total cost each week = 20x + 5y = T Example: Discussion Board Highlight

51. Adapted from Pearson Education, Inc. Copyright © 2009 Pearson Education, Inc. X = 5 Y = 3 20x + 5y = T Example: Discussion Board Highlight

52. Adapted from Pearson Education, Inc. Copyright © 2009 Pearson Education, Inc. At Action Water Sport of Ocean, City Maryland, the cost of renting a Jet ski is $42 per half hour, which includes a 5% sales tax. Determine the cost of a half hour Jet Ski rental before tax. Example

53. Adapted from Pearson Education, Inc. Copyright © 2009 Pearson Education, Inc. At Action Water Sport of Ocean, City Maryland, the cost of renting a Jet ski is $42 per half hour, which includes a 5% sales tax. Determine the cost of a half hour Jet Ski rental before tax. R = rental cost = ? tax = 5% = 0.05 T = total cost including tax = $42 R + (0.05)R = T Example

54. Adapted from Pearson Education, Inc. Copyright © 2009 Pearson Education, Inc. R + (0.05)R = T R + (0.05)R = 42 R(1+0.05) = 42 R(1.05) = 42 R = 40 The rental cost $40 per half hour before tax. Double Check: 40+(0.05)40 =? 42 Example – Jet Ski rental

55. Adapted from Pearson Education, Inc. Copyright © 2009 Pearson Education, Inc. Example Sandra Cone wants to fence in a rectangular region in her backyard for her lambs. She only has 184 feet of fencing to use for the perimeter of the region. What should the dimensions of the region be if she wants the length to be 8 feet greater than the width?

56. Adapted from Pearson Education, Inc. Copyright © 2009 Pearson Education, Inc. Example Sandra Cone wants to fence in a rectangular region in her backyard for her lambs. She only has 184 feet of fencing to use for the perimeter of the region. What should the dimensions of the region be if she wants the length to be 8 feet greater than the width? x + 8 x x = width of region x + 8 = length P = 2l + 2w 184 = 2(x) + 2 (x+8)

57. Adapted from Pearson Education, Inc. Copyright © 2009 Pearson Education, Inc. x = width of region x + 8 = length P = 2l + 2w x + 8 x Example - fencing

58. Adapted from Pearson Education, Inc. Copyright © 2009 Pearson Education, Inc. x = width of region x + 8 = length P = 2l + 2w x + 8 x The width of the region is 42 feet and the length is 50 feet. Example - fencing

59. Adapted from Pearson Education, Inc. Copyright © 2009 Pearson Education, Inc. Examples – Open Forum!

60. Adapted from Pearson Education, Inc. Copyright © 2009 Pearson Education, Inc. Thank You! Remember to Ask, Ask, Ask! kmeymaris@kaplan.edu AIM: kkmeymaris kmeymaris@kaplan.edu