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Math Help - Simultaneous Equations

  1. #1
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    Simultaneous Equations

    I am having trouble with the following simultaneous equations

    x = \alpha x + \beta y
    y = (1 - \alpha)x + (1 - \beta)y
    x + y = 1

    I also have the solutions:

     x = \frac{\beta}{1 + \beta  - \alpha} and  y  = \frac{1 -  \alpha}{1 + \beta - \alpha}

    but I don't know how these were worked out. Can someone please show me?
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  2. #2
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    Quote Originally Posted by MathTragic View Post
    I am having trouble with the following simultaneous equations

    x = \alpha x + \beta y
    y = (1 - \alpha)x + (1 - \beta)y
    x + y = 1

    I also have the solutions:

     x = \frac{\beta}{1 + \beta - \alpha} and  y = \frac{1 - \alpha}{1 + \beta - \alpha}

    but I don't know how these were worked out. Can someone please show me?
    first let  x = 1 - y

    sub this into y = (1 - \alpha)x + (1 - \beta)y

     y = (1 - \alpha)(1-y) + (1- \beta) y
     y = 1 - y - \alpha + \alpha y + y - y\beta
     y + y - \alpha y - y + y \beta = 1 - \alpha
     y (1 - \alpha + \beta) = 1 - \alpha
     y = \frac{1 - \alpha}{1 + \beta - \alpha}

    and then sub this into the first equation and solve for x
    Last edited by Katina88; September 15th 2009 at 06:33 PM.
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