The linear system of equations (LGS)
\begin{array}{rrr} A x + B y + C z & = & M \\
D y + E z & = & N \\
F z & = & L
\end{array}
has a unique solution
\begin{pmatrix} {\color{red}X} \\
{\color{blue}Y} \\ Z \end{pmatrix}.
Determine the entries
\color{red} X
=
X
\color{blue} Y
=
Y
Z
=
Z
In matrix notation, the linear system becomes
\begin{pmatrix}A & B & C & \bigl | &M \\
0 & D & E & \bigl | &N \\
0 & 0& F & \bigl | &L
\end{pmatrix} .
From the triangular form, we directly read from the third row/equation that
F z = L, thus Z = z = Z.
The second equation
D y + E z = N
can be rearranged to give D y = N - E z , so using that z = Z, we arrive at
{\color{blue}Y = y = Y}.
Using the values z = Z and {\color{blue} y = Y}
we found in the first equation
A x + B y + C z = M
leads to A {\color{red}x} = M - B {\color{blue}y} + C z , so we eventually find
{\color{red}X = x = X}.