I need to get, preferably by a numerical method, a solutions of: $$\left\{\begin{array}{lll} 2\sum_{i=1}^n b_{ik}x_i+x_{n+1}=0&\text{for}& k=1,2\ldots,n\\ \sum_{i=1}^n x_i=0 \end{array}\right.$$ with $(b_{ij})_{1\leq i,j\leq n}$ symmetric.
Is there a numerical method for homogeneous this problem?
Many thanks!