The basic idea of Gaussian elimination is to use the first equation to eliminate the first unknown x_1 from the remaining n-1 equations. Then we use the new second equation to eliminate the second unknown x_2 from the remaining n-2 equations. With n-1 such eliminations we obtain a so-called upper triangular set of equations of the form \begin{align} b_{11}x_1 +b_{12}x_2 +b_{13}x_3 + b_{14}x_4=&y_1 \nonumber \\ b_{22}x_2 + b_{23}x_3 + b_{24}x_4=&y_2 \nonumber \\ b_{33}x_3 + b_{34}x_4=&y_3 \nonumber \\ b_{44}x_4=&y_4. \nonumber \tag{11} \end{align} We can solve this system of equations recursively starting from x_n (in our case x_4 ) and proceed with what is called a backward substitution.