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 Multiplication and Division Properties of Equality: Equality: Let , , and represent algebraic expressions. Let , , and represent algebraic expressions. ͕ ͖ ͗ ͕ ͖ ͗ 1. Addition property of equality: 1. Multiplication property of equality: If , If , ͕ Ɣ ͖ ͕ Ɣ ͖ then then ͕ ƍ ͗ Ɣ ͖ ƍ ͗ ͕͗ Ɣ ͖͗ 2. Subtraction property of equality: 2. Division property of equality: If , If , ͕ Ɣ ͖ ͕ Ɣ ͖ then   ͕ Ǝ ͗ Ɣ ͖ Ǝ ͗ then (provided )  Ɣ  ͗ ƕ 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, , then the solution is all real numbers. This type of equation is called an identity . On the Ǝ13 Ɣ Ǝ13 other hand, if the variables are eliminated to reveal a false statement such as, , then there is no Ǝ7 Ɣ 3 solution . This type of equation is called a contradiction . All other linear equations which have only one solution are called conditional .

Examples: A. Check: ͬ ƍ 5 Ɣ Ǝ2

Ǝ5 Ǝ 5 ʚƎ7ʛ ƍ 5 Ɣ Ǝ2 ͬ Ɣ Ǝ7 (Solution Checks) Ǝ2 Ɣ Ǝ2 B. Check: ͬ Ǝ 3 Ɣ 7

ƍ3 ƍ3 10 Ǝ 3 Ɣ 7 (Solution Checks) ͬ Ɣ 10 7 Ɣ 7 C. Check: 3ͬ Ɣ 24

3ͬ 24 3ʚ8ʛ Ɣ 24 Ɣ 3 3 (Solution Checks) 24 Ɣ 24 ͬ Ɣ 8 D. Check: Ǝ7ͬ Ɣ 28

Ǝ7ͬ 28 Ǝ7ʚƎ4ʛ Ɣ 28 Ɣ Ǝ7 Ǝ7 (Solution Checks) 28 Ɣ 28 ͬ Ɣ Ǝ4 E. ͦ Check: ͧ ͬ Ɣ Ǝ4 ͧ ͦ ͯͨ ͧ (Multiply by the reciprocal) 2 Ǝ6 · ͬ Ɣ · ƴ Ƹ Ɣ Ǝ4 ͦ ͧ ͥ ͦ 3 1 (Solution Checks) ͬ Ɣ Ǝ6 Ǝ4 Ɣ Ǝ4 F. 4 Check: ͯͩ Ɣ 2 Ǝ10 Ǝ5 ͭ · Ɣ 2ʚƎ5ʛ Ɣ 2 1 Ǝ5 Ǝ5 (Solution Checks) ͭ Ɣ Ǝ10 2 Ɣ 2 Identity Example: Contradiction Example: G. H. 2ͬ ƍ 6 Ɣ 3ʚͬ ƍ 2ʛ Ǝ ͬ 5ͬ Ǝ 3 Ɣ 4ʚͬ ƍ 2ʛ ƍ ͬ

2ͬƍ6Ɣ3ͬƍ6Ǝͬ 5ͬƎ3Ɣ4ͬƍ8ƍͬ

2ͬ ƍ 6 Ɣ 2ͬ ƍ 6 5ͬ Ǝ 3 Ɣ 5ͬ ƍ 8

Ǝ2ͬ Ǝ 2ͬ Ǝ5ͬ Ǝ 5ͬ True Statement False Statement 6 Ɣ 6 Ǝ3 Ɣ 8 Solution: all real numbers No Solution I. 10 Ǝ 5ͬ Ɣ 3ʚͬ Ǝ 4ʛ Ǝ 2ʚͬ ƍ 7ʛ

10 Ǝ 5ͬ Ɣ 3ͬ Ǝ 12 Ǝ 2ͬ Ǝ 14 Check:

10 Ǝ 5ͬ Ɣ ͬ Ǝ 26 10 Ǝ 5ʚ6ʛ Ɣ 3ʚ6 Ǝ 4ʛ Ǝ 2ʚ6 ƍ 7ʛ

Ǝͬ Ǝ ͬ 10 Ǝ 30 Ɣ 3ʚ2ʛ Ǝ 2ʚ13 ʛ

10 Ǝ 6ͬ Ɣ Ǝ26 Ǝ20 Ɣ 6 Ǝ 26 (Solution Checks) Ǝ10 Ǝ 10 Ǝ20 Ɣ Ǝ20

Ǝ6ͬ Ɣ Ǝ36 ͯͪ3 ͯͧͪ ͯͪ Ɣ ͯͪ

ͬ Ɣ 6 J. 3ͯͦ 3ͯͨ (LCD is 10) K. Ǝ Ɣ 2 0.05ͬ ƍ 0.25 Ɣ 0.2 ͩ ͦ 100 ʚ0.05ͬ ƍ 0.25 ʛ Ɣ 0.2ʚ100 ʛ ͬ Ǝ 2 ͬ Ǝ 4 10 ƴ Ǝ Ƹ Ɣ 2ʚ10 ʛ 5ͬ ƍ 25 Ɣ 20 5 2 10 ʚͬ Ǝ 2ʛ 10 ʚͬ Ǝ 4ʛ Ǝ25 Ǝ 25 · Ǝ · Ɣ 20 ͩ3 ͯͩ 1 5 1 2 Ɣ ͩ ͩ 2ʚͬ Ǝ 2ʛ Ǝ 5ʚͬ Ǝ 4ʛ Ɣ 20 ͬ Ɣ Ǝ1 2ͬ Ǝ 4 Ǝ 5ͬ ƍ 20 Ɣ 20 Check:

Ǝ3ͬ ƍ 16 Ɣ 20 0.05 ʚƎ1ʛ ƍ 0.25 Ɣ 0.2 Ǝ16 Ǝ 16 Ǝ0.05 ƍ0.25 Ɣ 0.2 ͯͧ3 ͨ (Solution Checks) ͯͧ Ɣ ͯͧ 0.2 Ɣ 0.2 ͨ ͬ Ɣ Ǝ ͧ LINEAR EQUATIONS PRACTICE

1. ͦ ͧ 2 ͭ 4ͬ Ɣ 4 37. 68. ͬ ƍ ͨ Ɣ ͩ Ǝ ͩ ͬ Ɣ Ǝ45 2. ͬ ƍ 6 Ɣ Ǝ7 38. ͧ 69. ͨ ͬ Ǝ Ɣ Ǝ2ͬ Ǝ ͬ Ɣ Ǝ36 3. ͨ ͭ ͬ Ǝ 4 Ɣ 7 39. 3 3 ͥ 70. 3 4ͬ Ǝ 4 Ɣ Ǝ40 4. ͨ Ǝ ͪ Ɣ ͨ ͧ Ɣ Ǝ9 71. 40.   9ͬ Ǝ 7 Ɣ Ǝ34 5. ͕ Ǝ ƍ Ɣ 26 2ͬ ƍ 4 Ɣ 8 ͧ ͩ 72. ͫ ͭ Ǝ 6 Ɣ 8 6. 41. ͥͦ 5 ͬ 14 Ɣ 3 ƍ 2ͬ Ɣ ͥͤ ͦͩ 73. 7. 10 Ǝ ͬ Ɣ 6 8ͬ Ǝ 3 Ɣ Ǝ19 42. ͯͦ ͧ Ɣ 74. 8. ͪ ͭ Ǝ2ͬ Ǝ 4ͬ Ɣ 1 6 Ǝ ͬ Ɣ 9 43. 3ͮͨ ͧ 75. 9. Ɣ Ǝ9ͬ Ǝ 9ͬ Ɣ Ǝ9 Ǝͬ Ɣ Ǝ12 ͫ ͫ 76. 3 3 10. 44. ͨ3ͮͩ ͫ Ǝ Ɣ 2 3ʚͬ Ǝ 2ʛ Ɣ 6 ͪ Ɣ ͦ ͧ ͩ 11. 3 3 77. 3 3 Ǝ3ʚ2ͬ Ǝ 8ʛ Ɣ Ǝ12 45. Ǝ Ɣ 2 6 Ǝ ͨ Ɣ ͬ ͫ ͭ 12. 3 3 4ʚ6 ƍ 2ͬʛ Ɣ 0 46. 3 ͧ3 ͥ 78. Ǝ Ɣ ͧ Ǝ ͭ Ɣ 6 13. ͧ ͨ ͥͦ 3ͬ ƍ2ͬ ƍ6 Ɣ Ǝ15 79. 3 3 47. ͩ Ǝ Ɣ 1 14. Ǝ ͬ Ɣ 3ͬ ͬ ͭ 4 Ɣ Ǝ2ʚͬ ƍ 3ʛ ͦ ͩ ͧͯͩ4 ͦͯͨ4 80. 15. 48. ͭ ͭ Ǝ 4 Ɣ 6 27 Ɣ 46 ƍ 2ͬ Ǝ ͬ ͨ Ɣ ͧ 16. 81. 4ͬƍ6Ǝ7ͬƍ9 Ɣ 18 49. ͦ3ͯͥ ͧ3 ͩ ͬ ƍ 0.4ͬ Ɣ 3.5 Ǝ Ɣ ͧ ͨ ͪ 82. 17. 5ʚͬ Ǝ 3ʛ Ɣ 45 4 ƍ 3ʚͬ ƍ 2ʛ Ɣ 10 3 50. 18. Ǝ ͨ Ɣ 12 83. Ǝ3 ƍ 3ͬ Ɣ Ǝ2ʚͬ ƍ 1ʛ Ǝ3ʚͬ ƍ 7ʛ Ɣ 9 51. 19. Ǝͬ Ɣ Ǝ12 84. 9ͬ Ǝ6 Ɣ Ǝ3ͬ ƍ30 Ǝ4ʚͬ Ǝ 6ʛ Ɣ 12 52. 20. Ǝ2ͬ Ɣ Ǝ16 85. Ǝʚͬ ƍ 2ʛ Ɣ 2ʚ3ͬ Ǝ 6ʛ 8 Ɣ 2ʚͬ Ǝ 5ʛ ƍ 6ͬ 53. 21. 2ͬ Ɣ Ǝ14 86. 2ͬƍ6Ɣ3ͬƍ9Ǝ3 2 Ɣ 7ʚͬ ƍ 4ʛ ƍ 9ͬ ͥ 22. 54. 87. Ǝ5ͬ ƍ3 Ɣ 2ͬ ƍ10 ͫ ͬ Ɣ Ǝ8 1 Ɣ 3ʚͬ Ǝ 2ʛ ƍ 3 Ǝ 2ͬ 23. ͥ 88. 3ͬƎ12ͬ Ɣ24Ǝ9ͬ 55. 3 Ɣ 4ʚͬ Ǝ 2ʛ ƍ 5 Ǝ 3ͬ ͫ ͬ Ɣ 2 24. 89. 2ʚͬ ƍ 4ʛ Ɣ Ǝ3ʚͬ ƍ 5ʛ 56. 3 3.65 Ǝ 7.4ͬ ƍ 1.12 Ɣ 21.76 Ǝ Ɣ 4 ͦ 90. 25. Ǝ8ͬƍ3Ǝ2ͬ ƔƎ6ͬƍ3Ǝ4ͬ 4ʚ2ͬ Ǝ 3ʛ ƍ 4 Ɣ 8ͬ Ǝ 8 57. Ǝͬ Ɣ 26 91. 26. 10ͬƍ3ƍ10ͬ Ɣ13ͬƎ3ƍ7ͬ 6ͬ ƍ 11 Ɣ Ǝʚ6ͬ ƍ 5ʛ 58. 3ͬ Ɣ 15 92. 27. 6 ƍ 3ͬ Ɣ 5ʚͬ Ǝ 1ʛ Ǝ 3ʚͬ Ǝ 2ʛ 2ʚͬ ƍ 7ʛ Ɣ 6ͬ ƍ 9 Ǝ 4ͬ 59. 28. 4ͬ Ɣ Ǝ32 93. Ǝ5ʚ3 Ǝ 4ͬʛ Ɣ Ǝ6 ƍ 20ͬ Ǝ 9 10 Ǝ 5ͬ Ɣ 3ʚͬ Ǝ 4ʛ Ǝ 2ʚͬ ƍ 7ʛ 60. ͥ 29. ͬ Ɣ 5 94. 4ʚͬ Ǝ 3ʛ Ǝ ʚͬ Ǝ 5ʛ Ɣ 0 ͧ 9.2ͭ Ǝ4.3 Ɣ 50.9 30. 61. ͥ 95. Ǝ2ʚ4 Ǝ ͬʛ Ɣ 6ʚͬ ƍ 2ʛ ƍ 3ͬ ͭ ͬ Ɣ 5 0.05ͮ ƍ 0.2 Ɣ 0.15ͮ Ǝ 10.5 96. 31. ͨ 3 62. ͩ 0.25 ʚ60 ʛ ƍ 0.10ͬ Ɣ Ɣ Ǝ ͬ Ɣ Ǝ15 ͫ ͦͥ ͧ 0.15 ʚ60 ƍ ͬʛ 32. 3 ͯͦͤ 63. ͪ Ɣ Ǝ ͬ Ɣ Ǝ30 97. ͨ ͥͪ ͩ 0.5ʚ3ͥ ƍ 87 ʛ Ɣ 1.5ͥ ƍ 43.5 33. ͭ ͭ ͪ Ɣ 64. 98. ͥͤ ͩ ͩ ͬ Ɣ 90 0.4ʚͭ ƍ 10 ʛ ƍ 0.6ͭ Ɣ 2 ͥ 5ͮͥ 99. 34. 65. ͥ 21.1ͫ ƍ4.6 Ɣ 10.9ͫ ƍ 35.2 ͨ Ɣ ͨ ͬ Ɣ 4 ͧ 100. 35.  ͯͧ ͫ 0.125ͬ Ɣ 0.025 ʚ5ͬ ƍ 1ʛ Ɣ 66. ͩ ͦ ͪ ͬ Ɣ 168 36. ) ) ͥ Ɣ 9 Ǝ 67. ͥͤ ͩ ͪ ͬ Ɣ 2

LINEAR EQUATIONS PRACTICE ANSWERS

1. 26. ͨ 51. 77. ͬ Ɣ 1 ͬ Ɣ Ǝ ͬ Ɣ 12 ͬ Ɣ 63 2. ͧ 52. 78. ͬ Ɣ Ǝ13 27. No Solution ͬ Ɣ 8 ͬ Ɣ 27 3. 53. 79. ͬ Ɣ 11 28. All real numbers ͬ Ɣ Ǝ7 ͬ Ɣ 72 4. 54. 80. ͬ Ɣ Ǝ27 29. ͫ ͬ Ɣ Ǝ56 ͭ Ɣ 18 5. ͬ Ɣ 55. 81. ͬ Ɣ 2 ͧ ͬ Ɣ 14 ͬ Ɣ 2.5 6. ͥͥ 30. ͦͤ 56. 82. ͬ Ɣ ͬ Ɣ Ǝ ͫ ͬ Ɣ Ǝ8 ͬ Ɣ 12 ͦ 31. 57. 83. 7. ͬ Ɣ 12 ͬ Ɣ Ǝ26 ͬ Ɣ Ǝ10 ͬ Ɣ Ǝ2 58. 84. 8. 32. ͬ Ɣ 5 ͬ Ɣ 3 ͬ Ɣ Ǝ3 ͬ Ɣ Ǝ5 33. 59. 85. ͭ 9. ͗ Ɣ 2 ͬ Ɣ Ǝ8 ͬ Ɣ ͬ Ɣ 12 60. ͨ 10. 34. ͬ Ɣ 15 ͥͧ ͬ Ɣ 4 ͮ Ɣ 0 86. 35. 61. ͬ Ɣ Ǝ ͬ 11. ͕ Ɣ 5 ͬ Ɣ 45 ͬ Ɣ 6 62. 87. 12. 36. ͬ Ɣ 9 ͬ Ɣ 4 ͬ Ɣ Ǝ3 ͢ Ɣ 30 88. 37. 63. ͬ Ɣ 6 13. ͦͥ ͫ Ɣ 5 ͬ Ɣ 25 ͬ Ɣ Ǝ 64. 89. ͩ 38. ͥ ͬ Ɣ 75 ͬ Ɩ Ǝ2.3 14. ͬ Ɣ 90. All real numbers ͬ Ɣ Ǝ5 ͨ 65. 39. ͬ Ɣ 12 15. ͬ Ɣ 3 66. 91. No Solution ͬ Ɣ Ǝ19 ͬ Ɣ 144 16. 40. ͬ Ɣ Ǝ1 ͕ Ɣ 30 67. 92. 41. ͬ Ɣ 12 ͬ Ɣ Ǝ5 17. ͮ Ɣ 30 68. 93. ͬ Ɣ 0 ͬ Ɣ 25 ͬ Ɣ 6 ͥ 42. 94. 18. ͗ Ɣ Ǝ1 69. ͭ Ɣ 6 ͬ Ɣ ͩ 43. ͬ Ɣ 81 ͬ Ɣ Ǝ1 70. 95. 19. ͬ Ɣ Ǝ9 ͮ Ɣ 107 ͬ Ɣ 3 44. 96. 20. ͥͤ ͬ Ɣ 4 71. ͬ Ɣ 120 ͬ Ɣ 45. ͬ Ɣ Ǝ3 ͫ ͬ Ɣ 16 72. 97. All real numbers 21. 46. ͥ ͭ Ɣ 16 ͬ Ɣ 0 ͬ Ɣ Ǝ 73. 98. 22. ͩ ͬ Ɣ 4 ͭ Ɣ Ǝ2 ͬ Ɣ Ǝ1 ͩ ͥ 99. 47. 74. ͫ Ɣ 3 23. No Solution ͬ Ɣ ͬ ͬ Ɣ Ǝ ͪ 100. No Solution ͦͧ 48. 75. ͥ 24. ͭ Ɣ Ǝ1 ͬ Ɣ ͬ Ɣ Ǝ ͩ 49. ͦ 25. All real numbers ͬ Ɣ Ǝ14 76. 50. ͬ Ɣ 15 ͬ Ɣ Ǝ48