MATH

Solve the system of linear equations
2x+y-3z=4
4x+2z=10
-2x+3y-13z=-8

*im so lost (all these numbers are in one problem)

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  1. if you divide the 2nd by 2, all your equations start with 2x. That is good, since we could add or subtract them to eliminate the x

    2x+y-3z=4
    2x+z=10
    -2x+3y-13z=-8

    subtract 1st from the 2nd
    -y + 4z = 6 , #4

    add the 2nd and 3rd
    3y - 12z = 2 , #5

    #4 times 3---> -3y + 12z = 18
    #5 as is -----> 3y - 12z = 2

    add them: 0 = 20 , which is a contradiction
    AHHH, there is no unique solution.

    If you are studying equations of planes, you should categorize this as one of the special cases.

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    posted by Reiny
  2. well, the easy way is to google Gauss Jordan reduction calculator.
    http://www.gregthatcher.com/Mathematics/GaussJordan.aspx

    otherwise do it by elimination

    First eliminate z from first two:

    2x+y-3z = 4 times 2 = 4x+ 2y-6z = 8
    4x+0y+2z=10 times 3 = 12x+0y+6z = 30
    add the two
    16 x + 2 y = 38
    or
    8 x + y = 19

    Now eliminate z from the last 2
    multiply second by 13 and last by 2
    52 x + 0y + 26 z = 130
    -4 x + 6 y -26 z = -16
    ------------------------- add the two
    48 x + 6 y = 114
    or
    24 x + 3 y = 57
    Now you have the two equations in x and y
    8 x + 1 y = 19
    24 x + 3 y = 57

    multiply first by 3
    24 x + 3 y = 57
    Oh , my, system can not be solved, determinant is zero, two lines the same.

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    posted by Damon
  3. Your solution is illustrated by
    http://www.google.com/search?q=intersection+of+3+planes&tbm=isch&imgil=VKeuvId73BKgcM%253A%253Bhttp%253A%252F%252Ft2.gstatic.com%252Fimages%253Fq%253Dtbn%253AANd9GcTP5OK2B8xVEFseHgCh5S25ao2K4pX-fClEpupY9ltfPRK6cZC8NA%253B332%253B265%253BL5ht1c6dmTcU5M%253Bhttp%25253A%25252F%25252Fstweb.peelschools.org%25252Fjfsweb%25252Fmga4u%25252Funit3%25252Flesson7.html&source=iu&usg=__Epsu58j8aAIOLuG__2toxIf0TSk%3D&sa=X&ei=SN6gU9wz2KPIBJaOgPAK&sqi=2&ved=0CCkQ9QEwAg&biw=1680&bih=936#imgdii=_

    look at the 2nd picture, top left
    Your planes intersect in pairs, each one giving us a parallel line as the intersection of the planes, two at a time.

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    posted by Reiny

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