The new coefficients are $$ \begin{equation} b_{1k} = a_{1k}^{(1)} \quad k=1,\dots,n, \tag{14} \end{equation} $$ where each \( a_{1k}^{(1)} \) is equal to the original \( a_{1k} \) element. The other coefficients are $$ \begin{equation} a_{jk}^{(2)} = a_{jk}^{(1)}-\frac{a_{j1}^{(1)}a_{1k}^{(1)}}{a_{11}^{(1)}} \quad j,k=2,\dots,n, \tag{15} \end{equation} $$ with a new right-hand side given by $$ \begin{equation} y_{1}=w_1^{(1)}, \quad w_j^{(2)} =w_j^{(1)}-\frac{a_{j1}^{(1)}w_1^{(1)}}{a_{11}^{(1)}} \quad j=2,\dots,n. \tag{16} \end{equation} $$ We have also set \( w_1^{(1)}=w_1 \), the original vector element. We see that the system of unknowns \( x_1,\dots,x_n \) is transformed into an \( (n-1)\times (n-1) \) problem.