With the LU decomposition it is rather simple to solve a system of linear equations $$ \begin{align} a_{11}x_1 +a_{12}x_2 +a_{13}x_3 + a_{14}x_4=&w_1 \nonumber \\ a_{21}x_1 + a_{22}x_2 + a_{23}x_3 + a_{24}x_4=&w_2 \nonumber \\ a_{31}x_1 + a_{32}x_2 + a_{33}x_3 + a_{34}x_4=&w_3 \nonumber \\ a_{41}x_1 + a_{42}x_2 + a_{43}x_3 + a_{44}x_4=&w_4. \nonumber \end{align} $$
This can be written in matrix form as $$ \mathbf{Ax}=\mathbf{w}. $$
where \( \mathbf{A} \) and \( \mathbf{w} \) are known and we have to solve for \( \mathbf{x} \). Using the LU dcomposition we write $$ \mathbf{A} \mathbf{x} \equiv \mathbf{L} \mathbf{U} \mathbf{x} =\mathbf{w}. $$