Let $u_1,\ldots,u_n$ be a solution of an inhomogeneous system of linear equations. $$\begin{array}{ccl} \alpha_{11}x_1+\ldots+\alpha_{1n}x_n&=&\beta_1\\\ \alpha_{21}x_1+\ldots+\alpha_{2n}x_n&=&\beta_2\\ \vdots&\vdots&\vdots\\ \alpha_{m1}x_1+\ldots+\alpha_{mn}x_n&=&\beta_m.\\ \end{array}\quad\quad ( * )$$
Then all other solutions $w_1,\ldots,w_n$ of $( * )$ are given by
$$w_\nu:=u_\nu\pm h_\nu\quad\quad\text{ for all }\,\mu=1,\ldots,m,$$
where $h_1,\ldots,h_n$ is a solution of the corresponding homogeneous system of linear equations. $$\begin{array}{ccl} \alpha_{11}x_1+\ldots+\alpha_{1n}x_n&=&0\\\ \alpha_{21}x_1+\ldots+\alpha_{2n}x_n&=&0\\ \vdots&\vdots&\vdots\\ \alpha_{m1}x_1+\ldots+\alpha_{mn}x_n&=&0.\\ \end{array}$$
Proofs: 1
Applications: 1
