For our four-dimentional example this takes the form $$ \begin{align} y_1=&w_1 \nonumber\\ l_{21}y_1 + y_2=&w_2\nonumber \\ l_{31}y_1 + l_{32}y_2 + y_3 =&w_3\nonumber \\ l_{41}y_1 + l_{42}y_2 + l_{43}y_3 + y_4=&w_4. \nonumber \end{align} $$
and $$ \begin{align} u_{11}x_1 +u_{12}x_2 +u_{13}x_3 + u_{14}x_4=&y_1 \nonumber\\ u_{22}x_2 + u_{23}x_3 + u_{24}x_4=&y_2\nonumber \\ u_{33}x_3 + u_{34}x_4=&y_3\nonumber \\ u_{44}x_4=&y_4 \nonumber \end{align} $$
This example shows the basis for the algorithm needed to solve the set of \( n \) linear equations.