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System of linear equation question: Intersection of three equations

  1. Mar 10, 2009 #1
    1. The problem statement, all variables and given/known data

    Solve the system of the linear equations and interpret your solution geometrically:
    2x + y + 2z - 4 = 0 [1]
    x - y - z - 2 = 0 [2]
    x + 2y -6z - 12 = 0 [3]


    3. The attempt at a solution

    I've tried to eliminate the y variable:
    [1] + [2]
    3x + z - 6 = 0 [4]

    [1]x(-2) + [3]
    -3x -10z + 4 = 0 [5]

    Now solve for z
    [4] + [5]
    -9z - 2 = 0
    z = -2/9

    Is this correct so far?
    I'm not sure what to do now, do I do this whole process again and solve for another variable? Or can I sub the z into [4] or [5]?
     
  2. jcsd
  3. Mar 10, 2009 #2

    Mark44

    Staff: Mentor

    Yes, substitute your z value into equations 4 and 5, to solve for x. Now you know z and x, so substitute them into any of your first 3 equations.

    To check, make sure that all three of your starting equations are true statements when you replace x, y, and z with the values you have found. If all three equations are satisified, you're golden.
     
  4. Mar 10, 2009 #3
    Sub z into [4]
    3x + (-2/9) - 6 = 0
    x = 56/27

    Sub z into [5]
    -3x -10(-2/9) + 4
    x = 56/27

    So x = 56/27
    Now when I sub in the values x = 56/27 and z = -2/9 into [1] and [2] I get y = 8/27,
    but when I sub it into [3] I get 116/27.. Did I do something wrong?
     
  5. Mar 11, 2009 #4

    Mark44

    Staff: Mentor

    Yes.
    That + 4 should be -4.
    I get z = -10/9
     
  6. Mar 11, 2009 #5
    I don't really get it.

    [1] x (-2) = (-2)(2x + y + z - 4) = -4x -2y -4z +16
    + [3]
    -4x -2y - 4z +16 + x +2y -6z -12 = 0
    -3x -10z + 4 = 0

    16 - 12 = 4
     
  7. Mar 11, 2009 #6
    I've just tried to eliminate x first instead of y.. and I get totally different values for z -_-
     
  8. Mar 11, 2009 #7

    Mark44

    Staff: Mentor

    -2 * -4 = 8, not 16
     
  9. Mar 11, 2009 #8
    Oh geez. I must be blind. I have the correct answer now, thanks for your help.
     
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