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Mastering the Order of Operations in Algebra: A Comprehensive Guide

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This chapter covers the fundamental concepts of college algebra, specifically focusing on the order of operations, commonly remembered by the phrase "Please Excuse My Dear Aunt Sally" (PEMDAS). It highlights the importance of parentheses, exponents, multiplication, division, addition, and subtraction in mathematical expressions to ensure consistent results. Through worked examples, readers will learn how to evaluate various expressions correctly. The chapter also delves into evaluating algebraic expressions and simplifying complex expressions by combining like terms, providing a solid foundation for further algebra studies.

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Mastering the Order of Operations in Algebra: A Comprehensive Guide

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  1. Chapter 1 Review College Algebra

  2. Remember the phrase“Please Excuse My Dear Aunt Sally” or PEMDAS. ORDER OF OPERATIONS 1. Parentheses - ( ) or [ ] 2. Exponents or Powers 3. Multiply and Divide (from left to right) 4. Add and Subtract (from left to right)

  3. Evaluate 7 + 4 • 3. Is your answer 33 or 19? You can get 2 different answers depending on which operation you did first. We want everyone to get the same answer so we must follow the order of operations.

  4. Once again, evaluate 7 + 4  3 and use the order of operations. = 7 + 12 (Multiply.) =19(Add.)

  5. Example #114 ÷ 7 • 2 - 3 = 2 • 2 - 3 (Divide l/r.) = 4 - 3 (Multiply.) = 1(Subtract.)

  6. Example #23(3 + 7) 2 ÷ 5 = 3(10) 2 ÷ 5 (parentheses) = 3(100) ÷ 5 (exponents) = 300 ÷ 5 (multiplication) =60 (division)

  7. Example #320 - 3 • 6 + 102 + (6 + 1) • 4 = 20 - 3 • 6 + 102 + (7) • 4 (parentheses) = 20 - 3 • 6 + 100 + (7) • 4 (exponents) = 20 - 18 + 100 + (7) • 4 (Multiply l/r.) = 20 - 18 + 100 + 28 (Multiply l/r.) = 2 + 100 + 28 (Subtract l/r.) = 102 + 28 (Add l/r.) =130 (Add.)

  8. Answer Now Which of the following represents 112 + 18 - 33· 5in simplified form? • -3,236 • 4 • 107 • 16,996

  9. Answer Now Simplify16 - 2(10 - 3) • 2 • -7 • 12 • 98

  10. Answer Now Simplify24 – 6 · 4 ÷ 2 • 72 • 36 • 12 • 0

  11. Evaluating a Variable ExpressionTo evaluate a variable expression: • substitute the given numbers for each variable. • use order of operations to solve.

  12. Example # 4 n + (13 - n)  5 for n = 8 = 8 + (13 - 8)  5 (Substitute.) = 8 + 5 5 (parentheses) = 8 + 1 (Divide l/r.) = 9(Add l/r.)

  13. Example # 5 8y - 3x2+ 2n for x = 5, y = 2, n =3 = 8  2- 3  52 + 2  3 (Substitute.) = 8  2 - 3  25 + 2  3 (exponents) = 16- 3  25 + 2  3 (Multiply l/r.) = 16 -75 + 2  3 (Multiply l/r.) = 16 - 75 + 6 (Multiply l/r.) = -59 + 6 (Subtract l/r.) = -53 (Add l/r.)

  14. Answer Now What is the value of -10 – 4x if x = -13? • -62 • -42 • 42 • 52

  15. Answer Now What is the value of 5k3 if k = -4? • -8000 • -320 • -60 • 320

  16. Answer Now What is the value ofif n = -8, m = 4, and t = 2 ? • 10 • -10 • -6 • 6

  17. Evaluating Algebraic Expressions

  18. Expressions An expression is NOT an equation because it DOES NOT have an equal sign. There are 2 types of expressions. • 5 + 84 Numerical • 3x + 10 Algebraic Notice there are no equal signs in these expressions so they are not equations!

  19. Expressions A numerical expression contains only numbers and symbols and NO LETTERS. 5 times 3 plus 8 (5•3) + 8

  20. Expressions An algebraic expression contains only numbers, symbols, and variables. It is sometimes referred to as a variable expression. • The product of 3 and x 3x • The sum of m and 8 m + 8 • The difference of r and 2 r - 2

  21. Expressions What is a variable? A variable represents an unknown value. 2) 10 – ? 1) 4 + x 4) 20 3) 5y A variable can be any letter of the alphabet since it represents an unknown.

  22. Expressions Word Phrases for multiplication are: • The product of 5 and a number c • Seven times a number t • 6 multiplied by a number d 5 • c or 5c 7 • t or 7t 6 • d or 6d

  23. Expressions The Placement rule for multiplication is: • The product of 5 and a number c • Seven times a number t • 6 multiplied by a number d 5 • c or 5c • Always write the variable AFTER the number. 7 • t or 7t 6 • d or 6d

  24. Expressions Evaluate means to find the value of an algebraic expression by substituting numbers in for variables. m = 2 6 + m ? 6 + 2 8

  25. Expressions Evaluate means to find the value of an algebraic expression by substituting numbers in for variables. r = 3 7 + r ? 7 + 3 10

  26. Expressions Evaluate the variable expression when n = 6. 1) 2) Evaluate just means solve by substitution.

  27. Expressions Evaluate the variable expression when n = 6. 4) 3) Evaluate just means solve by substitution.

  28. Expressions Evaluate the algebraic expression when n = 8. 1) 2)

  29. Expressions Evaluate the expression if a = 3 and b = 4. 1) 2)

  30. Expressions Evaluate the expression if a = 3 and b = 4. 3) 4)

  31. Expressions Evaluate the expression if x = 5 and y = 3. 2) 1)

  32. Substitute & Evaluate #2 Evaluate ifx = 3 and y = 4 Evaluate means solve. Show the substitution. Show your work down. Circle your answer. Show your work one step at a time down. No equal signs!

  33. Combining Like Terms • In algebra we often get very long expressions, which we need to make simpler. Simpler expressions are easier to solve! • To simplify an expression we collect like terms. Like terms include letters that are the same and numbers.

  34. Let’s try one… • Step One: Write the expression. 4x + 5x -2 - 2x + 7 • Collect all the terms together which are alike. Remember that each term comes with an operation (+,-) which goes before it. 4x, 5x, and -2x -2 and 7 • Simplify the variable terms. 4x+5x-2x = 9x-2x = 7x • Simplify the constant (number) terms. -2+7 = 5 • You have a simplified expression by writing all of the results from simplifying. 7x + 5

  35. Another example… • 10x – 4y + 3x2 + 2x – 2y 3x2 10x, 2x -4y – 2y • 3x2+ 12x– 6y Remember you cannot combine terms with the same variable but different exponents.

  36. Now you try… Simplify the following: • 5x + 3y - 6x + 4y + 3z • 3b - 3a - 5c + 4b • 4ab – 2a2b + 5 – ab + ab2 + 2a2b + 4 • 5xy – 2yx + 7y + 3x – 4xy + 2x A A A A

  37. You Try #1 • Simplify the following: • 5x + 3y - 6x + 4y + 3z 5x, -6x 3y, 4y 3z -x+ 7y+ 3z

  38. You Try #2 • Simplify the following: • 3b - 3a - 5c + 4b 3b, 4b -3a -5c -3a+ 7b – 5c

  39. You Try #3 • Simplify the following: • 4ab – 2a2b + 5 – ab + ab2 + 2a2b + 4 4ab, -ab -2a2b, 2a2b 5, 4 ab2 3ab+ ab2+ 9

  40. You Try #4 • Simplify the following: • 5xy – 2yx + 7y + 3x – 4xy + 2x 5xy, -2yx, -4xy 7y 3x, 2x -xy+ 7y+ 5x

  41. Properties by Mr. Fitzgerald

  42. Distributive • Commutative • “order doesn’t matter” • Associative • “grouping doesn’t matter” • Identity properties of one and zero • Inverse “opposite” Five Properties

  43. Which Property?

  44. 1. Which Property? (2 + 1) + 4 = 2 + (1 + 4) Associative Property of Addition

  45. 2. Which Property? 3 + 7 = 7 + 3 Commutative Property of Addition

  46. 3. Which Property? 8 + 0 = 8 Identity Property of Addition

  47. 5. Which Property? 6 • 4 = 4 • 6 Commutative Property of Multiplication

  48. 6. Which Property? 17 + (-17) = 0 Inverse Property of Addition

  49. 7. Which Property? 2(5) = 5(2) Commutative Property of Multiplication

  50. 9. Which Property? 3(2 + 5) = 3•2 + 3•5 Distributive Property

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