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Math Help - solve an xor'd linear equation

  1. #1
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    solve an xor'd linear equation

    hello,
    i have 6 variables
    a=5
    b=10
    c=7
    d=9
    x=1
    y=2
    when i solve this equation using gauss-jordan it gives me wrong answer
    a⊕0⊕c⊕0⊕0⊕0=2
    0⊕b⊕0⊕d⊕0⊕0=3


    a⊕0⊕0⊕0⊕x⊕0=4
    0⊕b⊕0⊕0⊕0⊕y=8


    0⊕0⊕c⊕0⊕x⊕0=6
    0⊕0⊕0⊕d⊕0⊕y=11


    ⊕ is XOR


    thanks for your help
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  2. #2
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    Re: solve an xor'd linear equation

    The initial augmented matrix is
    Code:
    1 0 1 0 0 0 = 2
    0 1 0 1 0 0 = 3
    1 0 0 0 1 0 = 4
    0 1 0 0 0 1 = 8
    0 0 1 0 1 0 = 6
    0 0 0 1 0 1 = 11
    Replace row 3 by the sum of row 1 and row 3 (in the future, 3 := 1 + 3).

    Code:
    1 0 1 0 0 0 = 2
    0 1 0 1 0 0 = 3
    0 0 1 0 1 0 = 6
    0 1 0 0 0 1 = 8
    0 0 1 0 1 0 = 6
    0 0 0 1 0 1 = 11
    4 := 2 + 4

    Code:
    1 0 1 0 0 0 = 2
    0 1 0 1 0 0 = 3
    0 0 1 0 1 0 = 6
    0 0 0 1 0 1 = 11
    0 0 1 0 1 0 = 6
    0 0 0 1 0 1 = 11
    5 := 3 + 5

    Code:
    1 0 1 0 0 0 = 2
    0 1 0 1 0 0 = 3
    0 0 1 0 1 0 = 6
    0 0 0 1 0 1 = 11
    0 0 0 0 0 0 = 0
    0 0 0 1 0 1 = 11
    6 := 4 + 6

    Code:
    1 0 1 0 0 0 = 2
    0 1 0 1 0 0 = 3
    0 0 1 0 1 0 = 6
    0 0 0 1 0 1 = 11
    0 0 0 0 0 0 = 0
    0 0 0 0 0 0 = 0
    The last two equations are superfluous

    Code:
    1 0 1 0 0 0 = 2
    0 1 0 1 0 0 = 3
    0 0 1 0 1 0 = 6
    0 0 0 1 0 1 = 11
    2 := 4 + 2

    Code:
    1 0 1 0 0 0 = 2
    0 1 0 0 0 1 = 8
    0 0 1 0 1 0 = 6
    0 0 0 1 0 1 = 11
    1 := 3 + 1

    Code:
    1 0 0 0 1 0 = 4
    0 1 0 0 0 1 = 8
    0 0 1 0 1 0 = 6
    0 0 0 1 0 1 = 11
    This is a reduced row echelon form. Given the values of x and y we can find a, b, c and d. In particular, for x = 1 and y = 2,

    d + 2 = 11; d = 9
    c + 1 = 6; c = 7
    b + 2 = 8; b = 10
    a + 1 = 4; a = 5
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  3. #3
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    Re: solve an xor'd linear equation

    Thanks for the reply, emakarov
    but what if x and y werent given ?? what will i do ?
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  4. #4
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    Re: solve an xor'd linear equation

    This is an underdetermined system, so it has infinitely many solutions. For each value of x and y, there is a unique solution for a, b, c and d. For example, for x = 2 and y = 1 we have a = 6, b = 9, c = 4 and d = 10.
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  5. #5
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    Re: solve an xor'd linear equation

    Thank you so much emakarov! you are a life saver!
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