Consider the system \((E)\) of linear equations

\[ \begin{array}{rcccrcr} \colorbox{lightgreen}{$a_{11}$} {\color{red} x_1} & + & \cdots & + & \colorbox{lightgreen}{$a_{1n}$} {\color{red} x_n} & = & c_1 \\ \vdots\ \ \ & & & & \vdots\ \ \ & & \vdots\ \ \\ \colorbox{lightgreen}{$a_{m1}$} {\color{red} x_1} & + & \cdots & + & \colorbox{lightgreen}{$a_{mn}$} {\color{red} x_n} & = & c_m \end{array} \]

together with the associated system \((E_0)\) of homogeneous linear equations.

\[ \begin{array}{rcccrcr} \colorbox{lightgreen}{$a_{11}$} {\color{red} x_1} & + & \cdots & + & \colorbox{lightgreen}{$a_{1n}$} {\color{red} x_n} & = & 0 \\ \vdots\ \ \ & & & & \vdots\ \ \ & & \vdots\ \ \\ \colorbox{lightgreen}{$a_{m1}$} {\color{red} x_1} & + & \cdots & + & \colorbox{lightgreen}{$a_{mn}$} {\color{red} x_n} & = & 0 \end{array} \]

If \(\Vect{u}\) is a particular solution of \((E)\), then \(\Vect{x}=(x_1,\dots,x_n)\) solves \((E)\) if and only if \(\Vect{x} = \Vect{u} + \Vect{y}\) for some solution \(\Vect{y}\) of \((E_0)\).