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Math Help - Finding the general solution of a system of equations

  1. #1
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    Finding the general solution of a system of equations

    Find the general solution to the following system of equations and indicate which variables are free and which are basic.

    x_1 + 4x_4 + 3    = x_2 + x_3
    x_1 + 3x_4 + 1    = \frac{1}{2}x_3
    x_1 + x_2 + 2x_4 = 1

    Putting it in augmented matrix form to start we have:



    1 -1 -1 4 | -3
    1 0 -1/2 3 | -1
    1 1 0 2 | 1

    Now performing the following fundamental row operations:

    R1<-->R2
    R2+R3-->R2
    -2R3+R2-->R2
    -R3+R1-->R3
    R2/-2
    R2+R3-->R2
    -3R3+R1-->R1

    And finally I end with the augmented matrix:

    1 0 -2 0 | 5
    0 1 0 0 | 0
    0 0 -1/2 1 |-2

    Can someone please tell me if I got the correct matrix at the end and if so how do I determine which variables are free and which are basic?

    Thank you.
    Last edited by dumbledore; September 24th 2012 at 06:18 AM.
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  2. #2
    Senior Member MaxJasper's Avatar
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    Re: Finding the general solution of a system of equations

    Hints:
    For this this linear system to have a solution:
    x3 = 5-x1-3x2
    x4 = (1-x1-x2)/2

    Also try to solve using least squares when number of unknowns and equations are not equal...to obtain:

    {x1, x2, x3, x4} = {17/50, 69/50, 13/25, -9/25}
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  3. #3
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    Re: Finding the general solution of a system of equations

    I am not familiar with least squares. I presented this method because this is what I was taught and I need to know where I went wrong with the method I presented.
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