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Math Help - Simultaneous Equation Indices

  1. #1
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    Simultaneous Equation Indices

    Solve simultaneously:

    8^x / 4^y = 4
    11^(y-x) = 1/11
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  2. #2
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    Quote Originally Posted by acevipa View Post
    Solve simultaneously:

    8^x / 4^y = 4
    11^(y-x) = 1/11
    The first equation is readily simplified:

    \frac{8^x}{4^y} = 4 \Rightarrow 8^x = 4 \times 4^y = 4^{y + 1} \Rightarrow (2^3)^x = (2^2)^{y + 1} \Rightarrow 2^{3x} = 2^{2y + 2} \Rightarrow 3x = 2y + 2.

    The second is also readily simplified:

    11^{y - x} = \frac{1}{11} = 11^{-1} \Rightarrow y - x = -1.

    Solve the two simplified equations simultaneously .....
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  3. #3
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    Quote Originally Posted by acevipa View Post
    Solve simultaneously:

    8^x / 4^y = 4
    11^(y-x) = 1/11
    Use substitution method:

    8^x / 4^y = 4~\iff~8^x = 4^y \cdot 4 = 4^{y+1} .... [1]

    11^(y-x) = 1/11~\iff~ y = \frac1{121} + x .... [2]

    Plug in the term for y of [2] into [1]:

    8^x = 4^{\frac1{121} + x + 1} ~\iff~ 8^x = 4^{x+\frac{122}{121}} ~\iff~ 8^x = 4^x \cdot 4^{\frac{122}{121}} .... Now divide both sides by 4^x

    \frac{8^x}{4^x} = \left(\frac84\right)^x =  4^{\frac{122}{121}} ~\iff~ 2^x = (2^2)^{\frac{122}{121}}~\iff~ \boxed{2^x = 2^{\frac{244}{121}}}

    Therefore x = \frac{244}{121} .... and

    y = \frac{245}{121}
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  4. #4
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    Quote Originally Posted by earboth View Post
    Use substitution method:

    8^x / 4^y = 4~\iff~8^x = 4^y \cdot 4 = 4^{y+1} .... [1]

    11^(y-x) = 1/11~\iff~ y = \frac1{121} + x .... [2]

    Plug in the term for y of [2] into [1]:

    8^x = 4^{\frac1{121} + x + 1} ~\iff~ 8^x = 4^{x+\frac{122}{121}} ~\iff~ 8^x = 4^x \cdot 4^{\frac{122}{121}} .... Now divide both sides by 4^x

    \frac{8^x}{4^x} = \left(\frac84\right)^x =  4^{\frac{122}{121}} ~\iff~ 2^x = (2^2)^{\frac{122}{121}}~\iff~ \boxed{2^x = 2^{\frac{244}{121}}}

    Therefore x = \frac{244}{121} .... and

    y = \frac{245}{121}
    Sorry to say, earboth but I think you've done all that work for nothing .....

    If you look at the latex code used in the original post, you'll see [tex]11^(y-x) = 1/11[/tex].

    I think the intended latex code for equation 2 was [tex]11^{y-x} = 1/11[/tex], to give 11^{y-x} = 1/11 .....
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  5. #5
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    Quote Originally Posted by mr fantastic View Post
    Sorry to say, earboth but I think you've done all that work for nothing .....
    ( ................. ) here are included some forbidden words

    but nevertheles it was a nice training.
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