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Math Help - System of three equations and four variables

  1. #16
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    Re: System of three equations and four variables

    Well, you sure seem to know what you're doing.
    So if you can't solve this thing....
    I gather it is quite possible that there is a "short cut" staring us in the face!
    With the unwieldy equations that result in trying to solve the normal way,
    it seems that there MUST be a way to greatly simplify the process, else
    what teacher in his right mind would lay such a cruelty on a student?!
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  2. #17
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    Re: System of three equations and four variables

    If there is a real solution  \{x_1, x_2, x_3, x_4\} then \frac{x_1^2 x_4^2}{4} - (x_1 - x_4)^2 \ge 0 \Rightarrow \frac{x_1^2 x_4^2}{4} \ge (x_1 - x_4)^2. Try playing around with that inequality, as well as the other five or so you will obtain. Maybe you'll get something nice out of it...


    Clearly, x_1 = x_2 = x_3 = x_4 = 0 works. But is it the only solution?
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  3. #18
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    Re: System of three equations and four variables

    Quote Originally Posted by richard1234 View Post
    If there is a real solution  \{x_1, x_2, x_3, x_4\} then \frac{x_1^2 x_4^2}{4} - (x_1 - x_4)^2 \ge 0 \Rightarrow \frac{x_1^2 x_4^2}{4} \ge (x_1 - x_4)^2. Try playing around with that inequality, as well as the other five or so you will obtain. Maybe you'll get something nice out of it...
    I've already tried it (built the the locus of points) but it gives nothing interesting.
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  4. #19
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    Re: System of three equations and four variables

    Any new ideas, please?
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  5. #20
    Senior Member MaxJasper's Avatar
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    Re: System of three equations and four variables

    Quote Originally Posted by studenttt View Post
    How to solve this system?
    Attachment 24308
    Thanks a bunch!
    A simple solution is:
    Let x4=1 then: x1=x2=x3=0 = trivial solution
    Let x4=any number <>0 then x1=x2=x3=0 trivial solution.
    Let x4=0 then x1=0.72963, x2=x3=x4=0.

    Hence: the system only has solution when 3 of {x1,x2,x3,x4} be =0 at the same time!!
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  6. #21
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    Re: System of three equations and four variables

    Quote Originally Posted by MaxJasper View Post
    Hence: the system only has solution when 3 of {x1,x2,x3,x4} be =0 at the same time!!
    Hence? Why? Once more: the task is to find non-trivial roots, in particular, when all variables are nonzero. So far I don't see any solution.
    Last edited by studenttt; August 20th 2012 at 10:01 AM.
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  7. #22
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    Re: System of three equations and four variables

    Quote Originally Posted by MaxJasper View Post
    A simple solution is:
    Let x4=1 then: x1=x2=x3=0 = trivial solution
    Do you really think (0,0,0,1) is a root? We do not work with complex numbers.
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  8. #23
    Senior Member MaxJasper's Avatar
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    Re: System of three equations and four variables

    3 eqs 4 variables means you have infinite solutions for 3 variable while changing the 4th variable...therefore, all you get is 3 equations in terms of the 4th variable...with infinite values for 3 variables. However, this specific system has the solution I mentioned before.
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  9. #24
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    Re: System of three equations and four variables

    Quote Originally Posted by MaxJasper View Post
    A simple solution is:
    Let x4=1 then: x1=x2=x3=0 = trivial solution
    Let x4=any number <>0 then x1=x2=x3=0 trivial solution.
    Let x4=0 then x1=0.72963, x2=x3=x4=0.

    Hence: the system only has solution when 3 of {x1,x2,x3,x4} be =0 at the same time!!
    Can you prove that those are the only solutions?
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  10. #25
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    Re: System of three equations and four variables

    Quote Originally Posted by studenttt View Post
    Do you really think (0,0,0,1) is a root? We do not work with complex numbers.
    (0,0,0,1) is a root. Who said no complex numbers?
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  11. #26
    Senior Member MaxJasper's Avatar
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    Re: System of three equations and four variables

    I presented my proof by using math induction.
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