Solving Linear Equations
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The document provides information on solving linear equations, including the goal, methods, properties of equality, steps to solve equations, and examples. Specifically, it explains that the goal is to find the value of the variable that makes the statement true, and the method is to perform operations to isolate the variable using properties of equality like addition, subtraction, multiplication and division. It provides the steps to simplify, collect like terms, and check the solution, and gives examples of solving linear equations, identities, contradictions and conditional equations.
Solving Linear Equations
- 1. SOLVING LINEAR EQUATIONS Goal: The goal of solving a linear equation is to find the value of the variable that will make the statement (equation) true. Method: Perform operations to both sides of the equation in order to isolate the variable. Addition and Subtraction Properties of Equality: Let ܽ, ܾ, and ܿ represent algebraic expressions. 1. Addition property of equality: If ܽ ൌ ܾ, then ܽ ܿ ൌ ܾ ܿ 2. Subtraction property of equality: If ܽ ൌ ܾ, then ܽ െ ܿ ൌ ܾ െ ܿ Multiplication and Division Properties of Equality: Let ܽ, ܾ, and ܿ represent algebraic expressions. 1. Multiplication property of equality: If ܽ ൌ ܾ, then ܽܿ ൌ ܾܿ 2. Division property of equality: If ܽ ൌ ܾ, then ൌ (provided ܿ ് 0) Clearing Fractions or Decimals in an Equation: When solving an equation with fractions or decimals, there is an option of clearing the fractions or decimals in order to create a simpler equation involving whole numbers. 1. To clear fractions, multiply both sides of the equation (distributing to all terms) by the LCD of all the fractions. 2. To clear decimals, multiply both sides of the equation (distributing to all terms) by the lowest power of 10 that will make all decimals whole numbers. Steps for Solving a Linear Equation in One Variable: 1. Simplify both sides of the equation. 2. Use the addition or subtraction properties of equality to collect the variable terms on one side of the equation and the constant terms on the other. 3. Use the multiplication or division properties of equality to make the coefficient of the variable term equal to 1. 4. Check your answer by substituting your solution into the original equation. Note: If when solving an equation, the variables are eliminated to reveal a true statement such as, െ13 ൌ െ13, then the solution is all real numbers. This type of equation is called an identity. On the other hand, if the variables are eliminated to reveal a false statement such as, െ7 ൌ 3, then there is no solution. This type of equation is called a contradiction. All other linear equations which have only one solution are called conditional. Examples: A. ݔ 5 ൌ െ2 Check: ሺെ7ሻ 5 ൌ െ2 െ2 ൌ െ2 (Solution Checks) B. ݔ െ 3 ൌ 7 3 3 ݔ ൌ 10 Check: 10 െ 3 ൌ 7 7 ൌ 7 (Solution Checks) െ5 െ 5 ݔ ൌ െ7
- 2. C. 3ݔ ൌ 24 3ݔ 3 ൌ 24 3 ݔ ൌ 8 Check: 3ሺ8ሻ ൌ 24 24 ൌ 24 (Solution Checks) D. െ7ݔ ൌ 28 െ7ݔ െ7 ൌ 28 െ7 ݔ ൌ െ4 Check: െ7ሺെ4ሻ ൌ 28 28 ൌ 28 (Solution Checks) E. ଶ ଷ ݔ ൌ െ4 ଷ ଶ · ଶ ଷ ݔ ൌ ିସ ଵ · ଷ ଶ (Multiply by the reciprocal) ݔ ൌ െ6 Check: 2 3 ൬ െ6 1 ൰ ൌ െ4 െ4 ൌ െ4 (Solution Checks) F. ௬ ିହ ൌ 2 െ5 1 · ݕ െ5 ൌ 2ሺെ5ሻ ݕ ൌ െ10 Check: െ10 െ5 ൌ 2 2 ൌ 2 (Solution Checks) Identity Example: G. 2ݔ 6 ൌ 3ሺݔ 2ሻ െ ݔ 2ݔ 6 ൌ 3ݔ 6 െ ݔ 2ݔ 6 ൌ 2ݔ 6 െ2ݔ െ 2ݔ 6 ൌ 6 True Statement Solution: all real numbers Contradiction Example: H. 5ݔ െ 3 ൌ 4ሺݔ 2ሻ ݔ 5ݔ െ 3 ൌ 4ݔ 8 ݔ 5ݔ െ 3 ൌ 5ݔ 8 െ5ݔ െ 5ݔ െ3 ൌ 8 False Statement No Solution I. 10 െ 5ݔ ൌ 3ሺݔ െ 4ሻ െ 2ሺݔ 7ሻ 10 െ 5ݔ ൌ 3ݔ െ 12 െ 2ݔ െ 14 10 െ 5ݔ ൌ ݔ െ 26 െݔ െ ݔ 10 െ 6ݔ ൌ െ26 െ10 െ 10 െ6ݔ ൌ െ36 ି௫ ି ൌ ିଷ ି ݔ ൌ 6 Check: 10 െ 5ሺ6ሻ ൌ 3ሺ6 െ 4ሻ െ 2ሺ6 7ሻ 10 െ 30 ൌ 3ሺ2ሻ െ 2ሺ13ሻ െ20 ൌ 6 െ 26 െ20 ൌ െ20 (Solution Checks) J. ௫ିଶ ହ െ ௫ିସ ଶ ൌ 2 (LCD is 10) 10 ൬ ݔ െ 2 5 െ ݔ െ 4 2 ൰ ൌ 2ሺ10ሻ 10 1 · ሺݔ െ 2ሻ 5 െ 10 1 · ሺݔ െ 4ሻ 2 ൌ 20 2ሺݔ െ 2ሻ െ 5ሺݔ െ 4ሻ ൌ 20 2ݔ െ 4 െ 5ݔ 20 ൌ 20 െ3ݔ 16 ൌ 20 െ16 െ 16 ିଷ௫ ିଷ ൌ ସ ିଷ ݔ ൌ െ ସ ଷ K. 0.05ݔ 0.25 ൌ 0.2 100ሺ0.05ݔ 0.25ሻ ൌ 0.2ሺ100ሻ 5ݔ 25 ൌ 20 െ25 െ 25 ହ௫ ହ ൌ ିହ ହ ݔ ൌ െ1 Check: 0.05ሺെ1ሻ 0.25 ൌ 0.2 െ0.05 0.25 ൌ 0.2 0.2 ൌ 0.2 (Solution Checks)
- 3. LINEAR EQUATIONS PRACTICE 1. 4ݔ ൌ 4 2. ݔ 6 ൌ െ7 3. ݔ െ 4 ൌ 7 4. ௫ ଷ ൌ െ9 5. 2ݔ 4 ൌ 8 6. 14 ൌ 3 2ݔ 7. 8ݔ െ 3 ൌ െ19 8. 6 െ ݔ ൌ 9 9. െݔ ൌ െ12 10. 3ሺݔ െ 2ሻ ൌ 6 11. െ3ሺ2ݔ െ 8ሻ ൌ െ12 12. 4ሺ6 2ݔሻ ൌ 0 13. 3ݔ 2ݔ 6 ൌ െ15 14. 4 ൌ െ2ሺݔ 3ሻ 15. 27 ൌ 46 2ݔ െ ݔ 16. 4ݔ 6 െ 7ݔ 9 ൌ 18 17. 4 3ሺݔ 2ሻ ൌ 10 18. െ3 3ݔ ൌ െ2ሺݔ 1ሻ 19. 9ݔ െ 6 ൌ െ3ݔ 30 20. െሺݔ 2ሻ ൌ 2ሺ3ݔ െ 6ሻ 21. 2ݔ 6 ൌ 3ݔ 9 െ 3 22. െ5ݔ 3 ൌ 2ݔ 10 23. 3ݔ െ 12ݔ ൌ 24 െ 9ݔ 24. 2ሺݔ 4ሻ ൌ െ3ሺݔ 5ሻ 25. 4ሺ2ݔ െ 3ሻ 4 ൌ 8ݔ െ 8 26. 6ݔ 11 ൌ െሺ6ݔ 5ሻ 27. 2ሺݔ 7ሻ ൌ 6ݔ 9 െ 4ݔ 28. െ5ሺ3 െ 4ݔሻ ൌ െ6 20ݔ െ 9 29. 4ሺݔ െ 3ሻ െ ሺݔ െ 5ሻ ൌ 0 30. െ2ሺ4 െ ݔሻ ൌ 6ሺݔ 2ሻ 3ݔ 31. ସ ൌ ௫ ଶଵ 32. ௫ ସ ൌ ିଶ ଵ 33. ଽ ଵ ൌ ଽ ହ 34. ଵ ସ ൌ ௭ାଵ ସ 35. ହ ൌ ିଷ ଶ 36. ଵ ൌ 9 െ ହ 37. ଶ ଼ ଷ ସ ൌ ௪ ହ 38. ݔ െ ଷ ସ ൌ െ2ݔ 39. ௫ ସ െ ௫ ൌ ଵ ସ 40. ܽ െ ଷ ହ ൌ 26 41. ଵଶ ଵ ൌ ௭ ଶହ 42. ିଶ ൌ ଷ ଽ 43. ௫ାସ ൌ ଷ 44. ସ௫ାହ ൌ ଶ 45. 6 െ ௫ ସ ൌ ௫ ଼ 46. ௫ ଷ െ ଷ௫ ସ ൌ ଵ ଵଶ 47. ହ ଶ െ ݔ ൌ 3ݔ 48. ଷିହ௬ ସ ൌ ଶିସ௬ ଷ 49. ଶ௫ିଵ ଷ െ ଷ௫ ସ ൌ ହ 50. െ ௫ ସ ൌ 12 51. െݔ ൌ െ12 52. െ2ݔ ൌ െ16 53. 2ݔ ൌ െ14 54. ଵ ݔ ൌ െ8 55. ଵ ݔ ൌ 2 56. െ ௫ ଶ ൌ 4 57. െݔ ൌ 26 58. 3ݔ ൌ 15 59. 4ݔ ൌ െ32 60. ଵ ଷ ݔ ൌ 5 61. ଵ ଽ ݔ ൌ 5 62. െ ହ ଷ ݔ ൌ െ15 63. െ ହ ݔ ൌ െ30 64. ହ ݔ ൌ 90 65. ଵ ଷ ݔ ൌ 4 66. ݔ ൌ 168 67. ଵ ݔ ൌ 2 68. െ ଽ ହ ݔ ൌ െ45 69. െ ସ ଽ ݔ ൌ െ36 70. 4ݔ െ 4 ൌ െ40 71. 9ݔ െ 7 ൌ െ34 72. ଼ ݕ െ 6 ൌ 8 73. 10 െ ݔ ൌ 6 74. െ2ݔ െ 4ݔ ൌ 1 75. െ9ݔ െ 9ݔ ൌ െ9 76. ௫ ଷ െ ௫ ହ ൌ 2 77. ௫ െ ௫ ଽ ൌ 2 78. ௫ ଷ െ ௫ ଽ ൌ 6 79. ௫ ଼ െ ௫ ଽ ൌ 1 80. ହ ଽ ݕ െ 4 ൌ 6 81. ݔ 0.4ݔ ൌ 3.5 82. 5ሺݔ െ 3ሻ ൌ 45 83. െ3ሺݔ 7ሻ ൌ 9 84. െ4ሺݔ െ 6ሻ ൌ 12 85. 8 ൌ 2ሺݔ െ 5ሻ 6ݔ 86. 2 ൌ 7ሺݔ 4ሻ 9ݔ 87. 1 ൌ 3ሺݔ െ 2ሻ 3 െ 2ݔ 88. 3 ൌ 4ሺݔ െ 2ሻ 5 െ 3ݔ 89. 3.65 െ 7.4ݔ 1.12 ൌ 21.76 90. െ8ݔ 3 െ 2ݔ ൌ െ6ݔ 3 െ 4ݔ 91. 10ݔ 3 10ݔ ൌ 13ݔ െ 3 7ݔ 92. 6 3ݔ ൌ 5ሺݔ െ 1ሻ െ 3ሺݔ െ 2ሻ 93. 10 െ 5ݔ ൌ 3ሺݔ െ 4ሻ െ 2ሺݔ 7ሻ 94. 9.2ݕ െ 4.3 ൌ 50.9 95. 0.05ݖ 0.2 ൌ 0.15ݖ െ 10.5 96. 0.25ሺ60ሻ 0.10ݔ ൌ 0.15ሺ60 ݔሻ 97. 0.5ሺ3ݍ 87ሻ ൌ 1.5ݍ 43.5 98. 0.4ሺݕ 10ሻ 0.6ݕ ൌ 2 99. 21.1ݓ 4.6 ൌ 10.9ݓ 35.2 100. 0.125ݔ ൌ 0.025ሺ5ݔ 1ሻ
- 4. LINEAR EQUATIONS PRACTICE ANSWERS 1. ݔ ൌ 1 2. ݔ ൌ െ13 3. ݔ ൌ 11 4. ݔ ൌ െ27 5. ݔ ൌ 2 6. ݔ ൌ ଵଵ ଶ 7. ݔ ൌ െ2 8. ݔ ൌ െ3 9. ݔ ൌ 12 10. ݔ ൌ 4 11. ݔ ൌ 6 12. ݔ ൌ െ3 13. ݔ ൌ െ ଶଵ ହ 14. ݔ ൌ െ5 15. ݔ ൌ െ19 16. ݔ ൌ െ1 17. ݔ ൌ 0 18. ݔ ൌ ଵ ହ 19. ݔ ൌ 3 20. ݔ ൌ ଵ 21. ݔ ൌ 0 22. ݔ ൌ െ1 23. No Solution 24. ݔ ൌ െ ଶଷ ହ 25. All real numbers 26. ݔ ൌ െ ସ ଷ 27. No Solution 28. All real numbers 29. ݔ ൌ ଷ 30. ݔ ൌ െ ଶ 31. ݔ ൌ 12 32. ݔ ൌ െ5 33. ܿ ൌ 2 34. ݖ ൌ 0 35. ܽ ൌ 5 36. ݊ ൌ 30 37. ݓ ൌ 5 38. ݔ ൌ ଵ ସ 39. ݔ ൌ 3 40. ܽ ൌ 30 41. ݖ ൌ 30 42. ܿ ൌ െ1 43. ݔ ൌ െ1 44. ݔ ൌ 4 45. ݔ ൌ 16 46. ݔ ൌ െ ଵ ହ 47. ݔ ൌ ହ ଼ 48. ݕ ൌ െ1 49. ݔ ൌ െ14 50. ݔ ൌ െ48 51. ݔ ൌ 12 52. ݔ ൌ 8 53. ݔ ൌ െ7 54. ݔ ൌ െ56 55. ݔ ൌ 14 56. ݔ ൌ െ8 57. ݔ ൌ െ26 58. ݔ ൌ 5 59. ݔ ൌ െ8 60. ݔ ൌ 15 61. ݔ ൌ 45 62. ݔ ൌ 9 63. ݔ ൌ 25 64. ݔ ൌ 75 65. ݔ ൌ 12 66. ݔ ൌ 144 67. ݔ ൌ 12 68. ݔ ൌ 25 69. ݔ ൌ 81 70. ݔ ൌ െ9 71. ݔ ൌ െ3 72. ݕ ൌ 16 73. ݔ ൌ 4 74. ݔ ൌ െ ଵ 75. ݔ ൌ ଵ ଶ 76. ݔ ൌ 15 77. ݔ ൌ 63 78. ݔ ൌ 27 79. ݔ ൌ 72 80. ݕ ൌ 18 81. ݔ ൌ 2.5 82. ݔ ൌ 12 83. ݔ ൌ െ10 84. ݔ ൌ 3 85. ݔ ൌ ଽ ସ 86. ݔ ൌ െ ଵଷ ଼ 87. ݔ ൌ 4 88. ݔ ൌ 6 89. ݔ ൎ െ2.3 90. All real numbers 91. No Solution 92. ݔ ൌ െ5 93. ݔ ൌ 6 94. ݕ ൌ 6 95. ݖ ൌ 107 96. ݔ ൌ 120 97. All real numbers 98. ݕ ൌ െ2 99. ݓ ൌ 3 100. No Solution