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Math Help - Find x1,x2,x3,x4,x5

  1. #1
    Super Member dhiab's Avatar
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    Find x1,x2,x3,x4,x5

    Find the real numbers x_1 ,x_2 ,x_3 ,x_4 ,x_5 <br />
solutions this system, when m is the real parametr :

    \left\{ \begin{array}{l}<br />
 x_5  + x_2  = mx_1  \\ <br />
 x_1  + x_3  = mx_2  \\ <br />
 x_2  + x_4  = mx_3  \\ <br />
 x_3  + x_5  = mx_4  \\ <br />
 x_4  + x_1  = mx_5  \\ <br />
 \end{array} \right.
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  2. #2
    MHF Contributor
    Opalg's Avatar
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    Quote Originally Posted by dhiab View Post
    Find the real numbers x_1 ,x_2 ,x_3 ,x_4 ,x_5 <br />
solutions this system, when m is the real parametr :

    \left\{ \begin{array}{l}<br />
 x_5  + x_2  = mx_1  \\ <br />
 x_1  + x_3  = mx_2  \\ <br />
 x_2  + x_4  = mx_3  \\ <br />
 x_3  + x_5  = mx_4  \\ <br />
 x_4  + x_1  = mx_5  \\ <br />
 \end{array} \right.
    There is an obvious solution (x_1,x_2,x_3,x_4,x_5) = (0,0,0,0,0). For all but three values of m, that will be the only solution. The exceptional values of m are m=2 and m= \tfrac12(-1\pm\sqrt5).

    If m=2, the solution is (x_1,x_2,x_3,x_4,x_5) = s(1,1,1,1,1) (for any real number s).

    If m= \tfrac12(-1\pm\sqrt5) then m satisfies the equation m^2+m-1=0 and the solution is (x_1,x_2,x_3,x_4,x_5) = s(-1,-m,m,1,0) + t(m,-m,-1,0,1) (for any real numbers s and t).
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  3. #3
    Senior Member pacman's Avatar
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    dhiab, your problem post is getting harder . . . .
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  4. #4
    Super Member dhiab's Avatar
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    Quote Originally Posted by pacman View Post
    dhiab, your problem post is getting harder . . . .


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