# Thread: Find x1,x2,x3,x4,x5

1. ## Find x1,x2,x3,x4,x5

Find the real numbers $\displaystyle x_1 ,x_2 ,x_3 ,x_4 ,x_5 $$\displaystyle solutions this system, when m is the real parametr : \displaystyle \left\{ \begin{array}{l} x_5 + x_2 = mx_1 \\ x_1 + x_3 = mx_2 \\ x_2 + x_4 = mx_3 \\ x_3 + x_5 = mx_4 \\ x_4 + x_1 = mx_5 \\ \end{array} \right. 2. Originally Posted by dhiab Find the real numbers \displaystyle x_1 ,x_2 ,x_3 ,x_4 ,x_5$$\displaystyle$ solutions this system, when m is the real parametr :

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

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

If $\displaystyle m= \tfrac12(-1\pm\sqrt5)$ then m satisfies the equation $\displaystyle m^2+m-1=0$ and the solution is $\displaystyle (x_1,x_2,x_3,x_4,x_5) = s(-1,-m,m,1,0) + t(m,-m,-1,0,1)$ (for any real numbers s and t).

3. dhiab, your problem post is getting harder . . . .

4. Originally Posted by pacman
dhiab, your problem post is getting harder . . . .