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Math Help - prove in algebra

  1. #1
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    prove in algebra

    a/a-b = c/c-d
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  2. #2
    Super Member Bacterius's Avatar
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    Er ... and what are you trying to prove exactly ? Please add more detail ...
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  3. #3
    Member rowe's Avatar
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    Quote Originally Posted by the undertaker View Post
    a/a-b = c/c-d
    Here is an attempted proof, although it is probably wrong, I thought I'd have a go. It relies on some unwritten obvious proof, but you can prove that yourself.

    For a, b, c, d \in R, and a \neq b and c \neq d:

    Then \frac{a}{a-b}\cdot(a-b)\cdot(c-d) = \frac{c}{c-d}\cdot(a-b)\cdot(c-d)

    a(c-d) = c(a-b)

    \frac{c-d}{c} = \frac{a-b}{a}

    If x = \frac{1}{x} then x^2 = 1, and thus x = 1.

    Therefore, as shown above, \frac{a}{a-b} = \frac{a-b}{a} = 1, as does \frac{c}{c-d} = \frac{c-d}{c} = 1.

    1=1, QED.
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  4. #4
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    Quote Originally Posted by rowe View Post
    Here is an attempted proof, although it is probably wrong, I thought I'd have a go. It relies on some unwritten obvious proof, but you can prove that yourself.

    For a, b, c, d \in R, and a \neq b and c \neq d:

    Then \frac{a}{a-b}\cdot(a-b)\cdot(c-d) = \frac{c}{c-d}\cdot(a-b)\cdot(c-d)
    You assume here that \frac{a}{a-b}=\frac{c}{c-d} which is actually what you're trying to prove...

    a(c-d) = c(a-b)

    \frac{c-d}{c} = \frac{a-b}{a}
    For this step to be correct, you must assume that a,c are non-zero (if they are, the equation holds trivially, although I don't really know what we're trying to "prove" here...).

    If x = \frac{1}{x} then x^2 = 1, and thus x = 1.
    What about x=-1?

    Therefore, as shown above, \frac{a}{a-b} = \frac{a-b}{a} = 1, as does \frac{c}{c-d} = \frac{c-d}{c} = 1.

    1=1, QED.
    You basically said, "assume the equation holds, then it holds". This is obviously not a valid proof!

    Not to mention that it is quite obvious at a glance that this equation will definitely not hold for any four real numbers... what if we choose
    a=1, ~ b=2, ~ c=2, ~ d=1 ?
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  5. #5
    Member rowe's Avatar
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    Yeah, that was a pretty bad effort. Thanks for your pointers
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