\left\{\begin{matrix}
\:{a}_{11}{x}_{1}+{a}_{12}{x}_{2}+...+{a}_{1n}{x}_{n}={b}_{1}={a}_{1,n+1}\\
{a}_{21}{x}_{1}+{a}_{22}{x}_{2}+...+{a}_{2n}{x}_{n}={b}_{2}={a}_{2,n+1}\\
...................................................................\\
{a}_{n1}{x}_{1}+{a}_{n2}{x}_{2}+...+{a}_{nn}{x}_{n}={b}_{n}={a}_{n,n+1}
\end{matrix}\right.