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Can't find 'all' solutions

  1. Mar 30, 2005 #1
    I've been asked to find all integer solutions to the following equation.

    [tex]\frac{1}{x} + \frac{2}{y} - \frac{3}{z} = 1[/tex]

    Suppose I set y = 2, then it seems to me that there is an infinite number of solutions to the equation.

    Is there a systemic way for me to list ALL the integer solutions?
  2. jcsd
  3. Mar 30, 2005 #2

    matt grime

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    Rearrangements such as


    tell you that since x divides the rhs it divides yz, and so on, that may help with any systematic search
  4. Mar 30, 2005 #3


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    Since u have an equation with 3 unknowns,obviously the # of triplets/sollution is infinite in R.In N,things would go like that

    [tex] x=\frac{yz}{yz-2z+3y}\in \mathbb{N} [/tex]

  5. Mar 30, 2005 #4
    Is there a way for you to check the answer?
  6. Mar 30, 2005 #5
    I think you provided a pretty good argument. If you're trying to show that the solutions are infinite, assume one of the variables takes on on value (like you did with y=2), and say that there are an infinite number of x's and z's that solve the remaining equation.
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