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Math Help - Equation

  1. #1
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    Equation

    which points satisfies the problem below?


    XYZ+XZ+XY-YZ-11X-3Y-Z+9=0
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  2. #2
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    Quote Originally Posted by streethot View Post
    which points satisfies the problem below?


    XYZ+XZ+XY-YZ-11X-3Y-Z+9=0

    x = 0
    y = 0
    z = 9
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  3. #3
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    you have gave a particular solution... now, you have to prove that this is the only solution
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  4. #4
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    Quote Originally Posted by streethot View Post
    you have gave a particular solution... now, you have to prove that this is the only solution
    Lol! I'm not here to do your homework for you.

    You merely asked for a point that satisfied the equation, and I gave you one.

    If you want help with proving or otherwise that there's only one unique solution, then that's a different equation.
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  5. #5
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    Ah, how the road to mathematical truth is paved in blood...

    streethot:

    There are more solutions than you should care to count. A simple calculator program can tell you that...

    1,-1,-1
    1,-1,0
    1,-1,1
    -1,2,2
    0,2,1
    1,-1,-2
    1,-1,2
    -1,3,1
    0,1,3
    0,3,0
    etcetera ad nauseam

    As you know, Diophantine equations are notoriously difficult to solve. There may be an infinite number of solutions readily found by number-crunching.

    What exactly is it that you want to know about this equation? A proof of whether or not there is an infinitude of points? A formula generating a set of solutions?
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  6. #6
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    Thank you Media_Man. I was thinking that could have a way to solve this problem by the same method used to solve this:

    xyz+xy+yz+zx+x+y+z=243

    A classmate proposed this problem to me, i will investigate. Anyway, thanks for help me
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  7. #7
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    Here's a small part:

    If you rearrange the equation you'll get the following:

    Z=\frac{11X+3Y-XY-9}{(X-1)(Y+1)}

    When X=1, and Y=-1 we get Z=\frac{0}{0}, in other words 0 \cdot Z = 0

    So, any Z will work for X=1 and Y=-1


    Some other brute force stuff:
    When you set one variable to equal zero, there's only a finite number of solutions that will work (tried for X = 0 and Z = 0, Y = 0 will probably behave the same way)

    I don't think there's one general formula for the solution though, because of the (1,-1,Z) case ...
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  8. #8
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    Now i know what is the real question.

    Is for x,y,z\in\mathbb{N}^*
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  9. #9
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    Quote Originally Posted by streethot View Post
    Now i know what is the real question.

    Is for x,y,z\in\mathbb{N}^*
    Indeed, it is a diophantine equation.
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  10. #10
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    After much work, i got it! i will post with more calm after because the solution is a little long.
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