How to solve $x_j y_j = \sum_{i=1}^N x_i$ I have N equations and am having trouble with finding a solution.
$$\left\{\begin{matrix}
x_1 y_1 = \sum_{i=1}^N x_i\\
x_2 y_2 = \sum_{i=1}^N x_i\\
\vdots\\
x_N y_N = \sum_{i=1}^N x_i
\end{matrix}\right.$$
where $x_i$, ($i = 1, 2, \cdots, N$) is an unknown and $y_i$, ($i = 1, 2, \cdots, N$) is a known variable.
Given $y_i$'s, I have to find $x_i$'s but, I don't know where to start and even if it has a solution.

You can rewrite the system in the following form:
$$\left\{\begin{align*}
&(1-y_1)x_1+x_2+x_3+\ldots+x_N=0\\
&x_1+(1-y_2)x_2+x_3+\ldots+x_N=0\\
&x_1+x_2+(1-y_3)x_3+\ldots+x_N=0\\
&\qquad\qquad\qquad\qquad\vdots\\
&x_1+x_2+x_3+\ldots+(1-y_N)x_N=0\;,
\end{align*}\right.$$
or in matrix form as
$$\pmatrix{1-y_1&1&1&\dots&1&1\\1&1-y_2&1&\dots&1&1\\1&1&1-y_3&\dots&1&1\\\vdots&\vdots&\vdots&\ddots&\vdots&\vdots\\1&1&1&\dots&1-y_{N-1}&1\\1&1&1&\dots&1&1-y_N}\pmatrix{x_1\\x_2\\x_3\\\vdots\\x_{N-1}\\x_N}=\pmatrix{0\\0\\0\\\vdots\\0\\0}\;.$$
Solving this homogeneous linear system is in principle completely straightforward.
