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Thread: Simultaneous Equations

  1. #1
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    Simultaneous Equations

    I am having trouble with the following simultaneous equations

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

    I also have the solutions:

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

    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

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

    I also have the solutions:

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

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

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

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

    and then sub this into the first equation and solve for x
    Last edited by Katina88; Sep 15th 2009 at 05:33 PM.
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