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Is there any simpler methods to solve this?

  1. Jan 19, 2015 #1
    1. The problem statement, all variables and given/known data
    I was asked to solve this equation:
    ##{(x-\frac{1}{y})}^{2} -(y-\frac{1}{x})(x-\frac{1}{y})=9x##
    ##x-y=1##

    2. Relevant equations
    Simultaneous equations, factor theorem and quadratic formula

    3. The attempt at a solution
    I know I could have solved for x in the second equation, and substitute it into the first equation. However, the algebra becomes incredibly messy and the resulting equation becomes hard to solve. I see there is a very specific pattern in the first equation, though I can't really figure it out.
     
  2. jcsd
  3. Jan 19, 2015 #2

    ehild

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    Bring the terms in x-1/y and y-1/x to common denominator. Then factorize the left hand side.
     
  4. Jan 20, 2015 #3
    Now I have
    ##(\frac{xy-1}{y})(\frac{x^2 y-x-xy^2 +y}{xy})=9x##

    Then how should I proceed?

    Update: I finally managed to solve it. Thanks!
     
    Last edited: Jan 20, 2015
  5. Jan 20, 2015 #4

    ehild

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    How did you proceed? I thought of
    ##\frac{(xy-1)^2}{y^2} -\frac{xy-1}{x} \frac{xy-1}{y} = \frac{(xy-1)^2}{y} \left(\frac{1}{y}-\frac{1}{x}\right) ##, using x-y=1 substituting xy=u and solving for u first.
     
  6. Jan 20, 2015 #5
    I factored the equation, and substituted x-y for 1. Then I got (1-1/xy)(1-1/xy)=9, and (1-1/xy)^2 =9, and find an equation in terms of xy. Then I use the equation x-y=1 again to solve for x and y.
     
  7. Jan 20, 2015 #6

    ehild

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    Nice, it was what I had in my mind :)
     
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