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anonymous

  • one year ago

Medal Solve by determinants. x + y + z = 21 x=y +z = 3 x +y - z = 1

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  1. jim_thompson5910
    • one year ago
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    is there a typo in the second equation `x=y +z = 3` ??

  2. jim_thompson5910
    • one year ago
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    there are 2 equal signs there

  3. anonymous
    • one year ago
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    yes sorry its x - y + z = 3

  4. jim_thompson5910
    • one year ago
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    ok thanks

  5. jim_thompson5910
    • one year ago
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    were you able to set up the matrix?

  6. anonymous
    • one year ago
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    yes

  7. jim_thompson5910
    • one year ago
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    what matrix did you get

  8. anonymous
    • one year ago
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    1 1 1 1 -1 1 1 1 -1

  9. anonymous
    • one year ago
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    m i right?

  10. jim_thompson5910
    • one year ago
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    good

  11. jim_thompson5910
    • one year ago
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    that's your coefficient matrix

  12. jim_thompson5910
    • one year ago
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    do you know how to take the determinant of that matrix?

  13. anonymous
    • one year ago
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    no

  14. jim_thompson5910
    • one year ago
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    |dw:1443401707454:dw|

  15. jim_thompson5910
    • one year ago
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    focus on the first two columns |dw:1443401787922:dw|

  16. jim_thompson5910
    • one year ago
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    copy those columns and write it off to the right of the vertical bar |dw:1443401811957:dw|

  17. jim_thompson5910
    • one year ago
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    with me so far?

  18. anonymous
    • one year ago
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    yes

  19. jim_thompson5910
    • one year ago
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    ok so we have this extended matrix |dw:1443401923170:dw|

  20. jim_thompson5910
    • one year ago
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    circle the elements that fall along the diagonals like this |dw:1443401944912:dw| we have 3 subgroups

  21. anonymous
    • one year ago
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    ohh now i see it

  22. jim_thompson5910
    • one year ago
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    multiply the numbers in the subgroups |dw:1443402013404:dw|

  23. jim_thompson5910
    • one year ago
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    then add up the products: 1+1+1 = 3

  24. jim_thompson5910
    • one year ago
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    next, circle the elements along the diagonals like this |dw:1443402083998:dw|

  25. jim_thompson5910
    • one year ago
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    multiply out the numbers in each subgroup |dw:1443402112490:dw|

  26. jim_thompson5910
    • one year ago
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    then add up the products -1+1+(-1) = -1

  27. jim_thompson5910
    • one year ago
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    The first sum was 3 the second sum was -1 subtract the sums 3 - (-1) = 3+1 = 4

  28. jim_thompson5910
    • one year ago
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    so in the end, the determinant of that 3x3 matrix you wrote above is 4

  29. jim_thompson5910
    • one year ago
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    there are a lot of steps here, so go back over them if you aren't sure what's going on. And you can ask about any step

  30. anonymous
    • one year ago
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    so for x we got 3

  31. jim_thompson5910
    • one year ago
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    we haven't found x or y yet

  32. anonymous
    • one year ago
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    oh ok

  33. anonymous
    • one year ago
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    whats the next step

  34. jim_thompson5910
    • one year ago
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    do you see how I got the determinant to be 4?

  35. anonymous
    • one year ago
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    yes you added them

  36. jim_thompson5910
    • one year ago
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    so do you see how to find the determinant of any 3x3 matrix?

  37. anonymous
    • one year ago
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    yes i see it

  38. jim_thompson5910
    • one year ago
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    ok we'll let D = 4 be the determinant of the coefficient matrix

  39. jim_thompson5910
    • one year ago
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    if we replace the first column with the numbers 21, 3, 1 (found on the right side of each equation ) we get |dw:1443402459893:dw| what is the determinant of this new matrix (shown above)?

  40. anonymous
    • one year ago
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    1 1 1 1 -1 1

  41. anonymous
    • one year ago
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    we need to add another column

  42. jim_thompson5910
    • one year ago
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    did you start off doing this? |dw:1443402694898:dw|

  43. anonymous
    • one year ago
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    i got confused with the ones

  44. anonymous
    • one year ago
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    so i got ( 21 ) ( -1) ( -1) = 21

  45. anonymous
    • one year ago
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    (1) (1) (1) = 1 (1) (3) (1) = 2

  46. anonymous
    • one year ago
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    sorry 3*

  47. jim_thompson5910
    • one year ago
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    good, add up those products to get ??

  48. anonymous
    • one year ago
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    then you add them 21 + 1 + 3

  49. anonymous
    • one year ago
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    25

  50. jim_thompson5910
    • one year ago
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    ok keep that sum in mind now move onto the next set of diagonals

  51. anonymous
    • one year ago
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    how do you make that

  52. jim_thompson5910
    • one year ago
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    circling them helps me keep all the numbers straight |dw:1443402998519:dw|

  53. jim_thompson5910
    • one year ago
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    1*(-1)*1 = ?? 1*1*21 = ?? -1*3*1 = ?? then add up the products

  54. anonymous
    • one year ago
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    -1

  55. anonymous
    • one year ago
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    21

  56. anonymous
    • one year ago
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    -3

  57. anonymous
    • one year ago
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    add them 17

  58. anonymous
    • one year ago
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    m i right

  59. jim_thompson5910
    • one year ago
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    correct

  60. jim_thompson5910
    • one year ago
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    first sum = 25 second sum = 17 subtract the sums: first - second = 25 - 17 = 8

  61. jim_thompson5910
    • one year ago
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    So the determinant of this matrix (below) is 8 |dw:1443403306226:dw|

  62. jim_thompson5910
    • one year ago
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    since we replaced the first column, think of it as the x column, we call this matrix Dx

  63. jim_thompson5910
    • one year ago
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    to get the solution for x, we simply compute x = Dx/D

  64. jim_thompson5910
    • one year ago
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    Recall earlier we found D = 4 Dx = 8 so... x = Dx/D x = 8/4 x = 2

  65. anonymous
    • one year ago
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    and we do the same to find y and z

  66. jim_thompson5910
    • one year ago
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    yes you replace the second column with 23,3,1 and compute the determinant of that new matrix. That will give you Dy to find y, compute y = Dy/D

  67. jim_thompson5910
    • one year ago
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    same with z but you replace the third column with 23,3,1 z = Dz/D

  68. anonymous
    • one year ago
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    23 / 3 / 1

  69. jim_thompson5910
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    compute the determinant of this |dw:1443403528965:dw| to get Dy

  70. anonymous
    • one year ago
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    1 * 3 *1

  71. jim_thompson5910
    • one year ago
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    |dw:1443403666260:dw|

  72. jim_thompson5910
    • one year ago
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    |dw:1443403686878:dw|

  73. anonymous
    • one year ago
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    -3 23 1

  74. jim_thompson5910
    • one year ago
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    add those products up

  75. anonymous
    • one year ago
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    21

  76. jim_thompson5910
    • one year ago
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    good

  77. jim_thompson5910
    • one year ago
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    |dw:1443403784502:dw|

  78. anonymous
    • one year ago
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    3 1 -23

  79. jim_thompson5910
    • one year ago
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    add those values up

  80. anonymous
    • one year ago
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    -19

  81. jim_thompson5910
    • one year ago
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    first sum = 21 second sum = -19 first - second = 21 - (-19) = 21+19 = 40

  82. jim_thompson5910
    • one year ago
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    hold on, I think I made a typo somewhere

  83. jim_thompson5910
    • one year ago
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    oh my bad

  84. jim_thompson5910
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    I put 23 when I should have put 21

  85. jim_thompson5910
    • one year ago
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    |dw:1443404103846:dw| sorry about that

  86. anonymous
    • one year ago
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    3 1 -21

  87. anonymous
    • one year ago
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    -17

  88. jim_thompson5910
    • one year ago
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    |dw:1443404234817:dw|

  89. anonymous
    • one year ago
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    -3 21 1

  90. anonymous
    • one year ago
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    19

  91. jim_thompson5910
    • one year ago
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    |dw:1443404265866:dw|

  92. anonymous
    • one year ago
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    3 1 -21= -17

  93. jim_thompson5910
    • one year ago
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    so first sum - second sum = 19 - (-17) = 19+17 = 36 making Dy = 36

  94. jim_thompson5910
    • one year ago
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    y = Dy/D y = 36/4 y = 9

  95. anonymous
    • one year ago
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    now we have to do the same for z

  96. jim_thompson5910
    • one year ago
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    yes, replace the third column with 21,3,1 and compute the determinant

  97. anonymous
    • one year ago
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    1*3*21 = 63

  98. jim_thompson5910
    • one year ago
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    |dw:1443404399145:dw|

  99. jim_thompson5910
    • one year ago
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    |dw:1443404416172:dw|

  100. anonymous
    • one year ago
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    -21 3 1

  101. jim_thompson5910
    • one year ago
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    |dw:1443404436727:dw|

  102. anonymous
    • one year ago
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    -1 3 21

  103. anonymous
    • one year ago
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    23

  104. jim_thompson5910
    • one year ago
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    |dw:1443404481727:dw|

  105. anonymous
    • one year ago
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    -21 3 1

  106. anonymous
    • one year ago
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    -17

  107. jim_thompson5910
    • one year ago
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    first sum = 23 second sum = -17 first - second = ???

  108. anonymous
    • one year ago
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    40

  109. jim_thompson5910
    • one year ago
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    good

  110. jim_thompson5910
    • one year ago
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    Dz = 40, z = Dz/D = ???

  111. anonymous
    • one year ago
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    40 / what was d

  112. jim_thompson5910
    • one year ago
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    D = 4 we got this when we computed the determinant of the first matrix we set up (the coefficient matrix)

  113. anonymous
    • one year ago
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    10

  114. jim_thompson5910
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    z = 10, yes

  115. jim_thompson5910
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    x = 2 y = 9 z = 10

  116. jim_thompson5910
    • one year ago
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    I recommend you check it with each original equation in the system

  117. anonymous
    • one year ago
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    yes i will i will look at the steps again thankyou soo much for your time

  118. jim_thompson5910
    • one year ago
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    btw all of this is called cramer's rule http://www.purplemath.com/modules/cramers.htm

  119. anonymous
    • one year ago
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    oh ok thank you again :)

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