Gaussian Elimination
This process can be expressed mathematically as
$$
\begin{equation}
x_m = \frac{1}{b_{mm}}\left(y_m-\sum_{k=m+1}^nb_{mk}x_k\right)\quad m=n-1,n-2,\dots,1.
\tag{12}
\end{equation}
$$
To arrive at such an upper triangular system of equations, we start by eliminating
the unknown \( x_1 \) for \( j=2,n \). We achieve this by multiplying the first equation by \( a_{j1}/a_{11} \) and then subtract
the result from the $j$th equation. We assume obviously that \( a_{11}\ne 0 \) and that
\( \mathbf{A} \) is not singular.