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Math Help - Transposition

  1. #1
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    Transposition

    Hi could someone please tell me if this is correct, and steer me in the right direction if it isn't?


    Transpose the following for r:

    x=\frac{1}{r-1}-\frac{1}{r+1}+1


    x-1=\frac{1}{r-1}-\frac{1}{r+1}

    r-1=\frac{1}{x-1}-\frac{1}{r+1}

    (r-1)(r+1)=\frac{1}{x-1}-1

    r^2-1=\frac{1}{x-1}-1

    r^2=\frac{1}{x-1}

    r=\sqrt\frac{1}{x-1}

    Thanks in advance!
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  2. #2
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    Quote Originally Posted by andyw View Post
    x=\frac{1}{r-1}-\frac{1}{r+1}+1
    x-1=\frac{1}{r-1}-\frac{1}{r+1}
    r-1=\frac{1}{x-1}-\frac{1}{r+1}\;\;\;{\color{red}This\;is\;incorrect  ...}
    (r-1)(r+1)=\frac{1}{x-1}-1\;\;\;{\color{red}And\;so\;is\;this...}
    ...
    You want to solve for r, right?
    \begin{aligned}<br />
x &= \frac{1}{r - 1} - \frac{1}{r + 1} + 1 \\<br />
x - 1 &= \frac{1}{r - 1} - \frac{1}{r + 1} \\<br />
(x - 1){\color{red}(r^2 - 1)} &= \left(\frac{1}{r - 1} - \frac{1}{r + 1}\right){\color{red}(r^2 - 1)} \\<br />
(x - 1)(r^2 - 1) &= r + 1 - (r - 1) \\<br />
xr^2 - x - r^2 + 1 &= r + 1 - r + 1<br />
\end{aligned}

    \begin{aligned}<br />
xr^2 - r^2 - x + 1 &= 2 \\<br />
xr^2 - r^2 &= x + 1 \\<br />
r^2(x - 1) &= x + 1 \\<br />
r^2 &= \frac{x + 1}{x - 1} \\<br />
r &= \sqrt{\frac{x + 1}{x - 1}} \\<br />
\end{aligned}


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  3. #3
    MHF Contributor Bruno J.'s Avatar
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    x=\frac{1}{r-1}-\frac{1}{r+1}+1<br />

    = \frac{r+1-(r-1)+r^2-1}{r^2-1} = \frac{r^2+1}{r^2-1}


    x(r^2-1)= r^2+1

    r^2(x-1)=x+1

    r=\sqrt{\frac{x+1}{x-1}}
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  4. #4
    MHF Contributor Bruno J.'s Avatar
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    Quote Originally Posted by yeongil View Post
    You want to solve for r, right?
    <br />
xr^2 - r^2 - x + 1 = 2<br />
    <br />
xr^2 - r^2 = x - 1<br />
    this step is incorrect.

    There was no equation to solve by the way. Solution to what? If I set r=2 then x=5/3. Isn't that a solution?
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  5. #5
    MHF Contributor Bruno J.'s Avatar
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    Caught it just in time!
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  6. #6
    Senior Member Stroodle's Avatar
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    r=\pm\sqrt{\frac{x+1}{x-1}}
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