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Math Help - Rearrange an Algebraic Equation

  1. #1
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    Rearrange an Algebraic Equation

    I would like to rearrange the equation:
    X = (Y-A)/(Y-B) to look like this
    B-A = ((X-1)*(Y-A))/X

    I'm able to isolate B-A on the left hand side of the equation, but I
    can't seem to get the right hand side to the form above.

    I've tried the following steps:
    1. XY-XB = Y-A
    2. XY-XB-Y = -A
    3. XY-Y = XB-A
    4. XY-Y = X(B-(A/X))
    5. (XY-Y)/X = B-(A/X)
    6. B = (XY-Y)/X + (A/X) = (XY-Y+A)/X
    7. B-A = ((XY-Y+A)/X)-A

    and this is where I get stuck. I feel like I'm close but I can't
    quite get it in the form shown above. What am I missing?
    Thanks.
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  2. #2
    A Plied Mathematician
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    If the equation you're trying to prove is indeed true, given the original equation, I would do the following: solve for B and call that Equation 1; then solve for A and call that Equation 2; subtract the latter from the former, and simplify.

    [EDIT]: This approach doesn't work. See next post.
    Last edited by Ackbeet; July 9th 2010 at 09:43 AM. Reason: Approach not usable.
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  3. #3
    A Plied Mathematician
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    Ok, the equation works. I went backwards from the equation you're trying to prove to the equation you started with. What I would do is look at the process used to do that reverse proof, and then reverse it again. All the steps are reversible. In other words, show that by assuming

    B-A=\frac{(X-1)(Y-A)}{X},

    you can derive

    X=\frac{Y-A}{Y-B}.

    Then simply reverse your proof to get your desired result.
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  4. #4
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    earboth's Avatar
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    Quote Originally Posted by triviaguy View Post
    I would like to rearrange the equation:
    X = (Y-A)/(Y-B) to look like this
    B-A = ((X-1)*(Y-A))/X

    I'm able to isolate B-A on the left hand side of the equation, but I
    can't seem to get the right hand side to the form above.

    I've tried the following steps:
    1. XY-XB = Y-A
    2. XY-XB-Y = -A
    3. XY-Y = XB-A
    4. XY-Y = X(B-(A/X))
    5. (XY-Y)/X = B-(A/X)
    6. B = (XY-Y)/X + (A/X) = (XY-Y+A)/X
    7. B-A = ((XY-Y+A)/X)-A

    and this is where I get stuck. I feel like I'm close but I can't
    quite get it in the form shown above. What am I missing?
    Thanks.
    All your calculations are correct. You only stopped 2 steps in front of the finish. I'll take your last line:

    B-A = \left(\dfrac{XY-Y+A}{X}\right) - A

    B-A = \dfrac{XY-Y+A-AX}{X}

    B-A = \dfrac{Y(X-1)-A(X-1)}{X}

    B-A = \dfrac{(Y-A)(X-1)}{X}
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  5. #5
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    Thanks for the input Ackbeet, I believe I've come up with the correct procedure to rearrange the original equation as follows:

    X = (Y-A)/(Y-B)

    1. XY-XB = Y-A
    2. XY-Y = XB-A
    3. Y(X-1) = XB-A
    4. Y(X-1)-XA = XB-A-XA
    5. Y(X-1)-XA+A = XB-XA
    6. (Y(X-1)-XA+A)/X = B-A
    7. (Y(X-1)-A(X-1))/X = B-A
    8. ((X-1)*(Y-A))/X = B-A
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  6. #6
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    Thanks earboth, I found the solution as well. It's funny how you sometimes can't see something when it's right in front of you. Thanks again to all who responded.
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