How to solve for any given natural number n?

Question

I started with hard way of putting the coefficients into a matrix. But, iz did not help.

the following system of linear equations: system

Answer 1

Consider the sum of equations : $$(n-1)(x_{1}+..+x_{n}) = \frac{n(n+1)}{2}$$

Now you could find $$x_{i} = \frac{n(n+1)}{2(n-1)} - i$$.

Answer 2

Well, by adding your equations we have $(n-1)(x_1+...+x_n)=1+...+n=\frac{n(n+1)}{2}$. And therefore $x_1+...+x_n=\frac{n(n+1)}{2(n-1)}$. Now, by the $i$-th equation, $x_1+...+x_n-x_i=i$ so $\frac{n(n+1)}{2(n-1)}-x_i=i$. Finally, $x_i=\frac{n(n+1)}{2(n-1)}-i$ $ \forall i \in \{1,...,n \} $