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Looking for an elegant proof

  1. Aug 5, 2011 #1
    Given the simultaneous equations for real numbers x and y:

    [itex]\sqrt{x}+y=7[/itex]

    and

    [itex]\sqrt{y}+x=11[/itex]

    Find the solution. Guessing it is easy (the answers are 9 and 4) and the brute force way to do it is when you square and make subsitutions, ultimately leading to an equation of the fourth power in one variable.

    Is there a more elegant but formal way, that doesn't require me to guess the answer?
     
  2. jcsd
  3. Aug 5, 2011 #2
    You can do it by substituting in each equation then solving for x or y. You still have to guess 9, but it gives you the other roots too.

    [tex]x=(7-y)^2[/tex]
    [tex]y=(11-x)^2[/tex]
    After substitution I got:
    [tex]x^4-44x^3+712x^2-5017x+12996=0[/tex]
    Factorizing out x=9
    [tex](x-9)(x^3-35x^2+397x-1444)=0[/tex]
    Then you can use the cubic root formula:
    http://en.wikipedia.org/wiki/Cubic_function#General_formula_of_roots
    But instead I used:
    http://www.wolframalpha.com/input/?i=root+of+x^4-44x^3+712x^2-5017x+12996
     
  4. Aug 6, 2011 #3
    Maybe letting $a=\sqrt{x}$ and $b=\sqrt{y}$? I haven't tried it yet, but it might work.
     
  5. Aug 6, 2011 #4
    There's the rational root theorem that'll give you all rational solutions to a polynomial.
     
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