Solving_Linear_Equations.pdf

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
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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.   5  2 5 5   7 B.   3  7 3 3   10 Check: 7  5  2 2  2 (Solution Checks) Check: 10  3  7 7  7 (Solution Checks) C. 3  24 3 24  3 3 8 D. 7  28 7 28  7 7   4 E.     F.   4   ·   !  Check: 38  24 24  24 (Solution Checks) Check: 74  28 28  28 (Solution Checks)  · (Multiply by the reciprocal)   6  2 5 " ·  25 1 5 "  10 Identity Example: G. 2  6  3  2   2  6  3  6   2  6  2  6  2 2 6  6 True Statement Solution: all real numbers I. 10  5  3  4  2  7 10  5  3  12  2  14 10  5    26    10  6  26 10  10 6  36 #$ # J. $ !    $  Check: 10  56  36  4  26  7 10  30  32  213 20  6  26 20  20 (Solution Checks) #  2 (LCD is 10) 2 4    210 5 2 10   2 10   4 ·  ·  20 1 5 1 2 2  2  5  4  20 2  4  5  20  20 3  16  20 16  16 10  Contradiction Example: H. 5  3  4  2   5  3  4  8   5  3  5  8  5 5 3  8 False Statement No Solution # 6 $ Check: 2 6    4 3 1 4  4 (Solution Checks) Check: 10 2 5 2  2 (Solution Checks)        K. 0.05  0.25  0.2 1000.05  0.25  0.2100 5  25  20 25  25 !$ !  ! !   1 Check: 0.051  0.25  0.2 0.05  0.25  0.2 0.2  0.2 (Solution Checks) LINEAR EQUATIONS PRACTICE 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 4  4   6  7 47 $   9 2  4  8 14  3  2 8  3  19 6  9   12 3  2  6 32  8  12 46  2  0 3  2  6  15 4  2  3 27  46  2   4  6  7  9  18 4  3  2  10 38. 39. 40.      2 69.    36 $ 4  4  40   $  70.   71.   #  ( 42.  #  43. $+ 47. 
   26  46. !  41. 45. ' # * 73. ) 74.   '  75. '  $ 6  $  $ $ -      3     !  48. !    49. $   $ $    9  6  3  30 51.   12 2  6  3  9  3 53.   2  23  6 5  3  2  10 3  12  24  9 2  4  3  5 52. 2  14 53  4  6  20  9 59. 3  15 2  7  6  9  4 4  3    5  0 60.   5 61.  ) 5 '  32. $   33. ) ( $    15 # 63.    30 ! ( ) 82. 83. 84. 85. 86. 89. 90. 91. 92. 93. 94. 95. 96.  ! 97.   90 98.  65.   4 99.  66. ' #   168 67.  # 2   *+ 35.  !    9 , ! !  '  $  2 ! $  2 ) $  6 ) $ -  1 ) "46 )   0.4  3.5 5  3  45 3  7  9 4  6  12 8  2  5  6 2  7  4  9 1  3  2  3  2 3  4  2  5  3 3.65  7.4  1.12  21.76 8  3  2  6  3  4 10  3  10  13  3  7 6  3  5  1  3  2 10  5  3  4  2  7 9.2"  4.3  50.9 0.05/  0.2  0.15/  10.5 0.2560  0.10  0.1560   # 64.  , 80. # 34. ( ! 62.   4  32 9  9  9 ! 88. 58. 2  4  1 79. 81. "68 10    6 $ 2 6  11  6  5 - 78.   ' $ 55.   26 9  7  34 77. 87. $ ) $   8 '  76. 54. ' ! $  57.  # 2  16  4 24    6  2  3 !  56. 42  3  4  8  8 72. !  $+! !   12 3  3  2  1 )    45     44.. 68. - 50. 31. 36. 37. 0.530  87  1.50  43.5 0.4"  10  0.6"  2 21.11  4.6  10.91  35.2 100. 0.125  0.0255  1 LINEAR EQUATIONS PRACTICE ANSWERS 1. 2. 3. 4. 5.      1  13  11  27 2 6.     7.   2 8.   3 9.   12 10.   4 11.   6 12.   3 13.    14. 15. 16. 17.      !  5  19  1 0  18.   ! 19.   3 20.   ( ' 21.   0 22.   1 23. No Solution 24.     ! 25. All real numbers 26.      27. No Solution 28. All real numbers 29.   '  30.    31. 32. 33. 34. 35. 36. 37. ( '   12   5 2 /0
5 2  30 15  38.    39. 40. 41. 42. 43. 44. 45. 3
 30 /  30   1   1 4   16  46.    ! ! 47.   - 48. "  1 49.   14 50.   48 51. 52. 53. 54. 55. 56. 57. 58. 59. 60. 61. 62. 63. 64. 65. 66. 67. 68. 69. 70. 71. 72. 73.   12 8   7   56   14   8   26 5   8   15   45 9   25   75   12   144   12   25   81   9   3 "  16 4  74.    #  75.    76.   15 77. 78. 79. 80. 81. 82. 83. 84.   63   27   72 "  18   2.5   12   10 3 85.   )  86.     - 87.   4 88.   6 89.  3 2.3 90. All real numbers 91. No Solution 92.   5 93.   6 94. "  6 95. /  107 96.   120 97. All real numbers 98. "  2 99. 1  3 100. No Solution