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.

We have that for each $j$
$$
x_j(1-y_j) + \sum_{i=1}^{j-1} x_i + \sum_{i=j+1}^{N} x_i = 0
$$
Given that the $y_i$'s are known, we can just use Gaussian elimination to solve this.
