Let \(F\) be a field (e.g. the field of rational numbers $\mathbb Q$, the real numbers $\mathbb R$, or the complex numbers $\mathbb C$) and let \(\alpha_1,\ldots,\alpha_n,\gamma \in F\). A linear equation with with \(n\) variables (or unknowns) \(x_1,\ldots,x_n\) is the equation \[\alpha_1x_1+\ldots+\alpha_nx_n=\beta\quad\quad ( * ).\] The linear equation is called * homogeneous, if $\beta= 0$ and * inhomogeneus, if $\beta\neq 0.$
We say that the linear equation has a solution, if there exist numbers \(a_1,\ldots,a_n\in F\) such that for the replacement \(x_1:=a_1,\ldots,x_n:=a_n\) the equation $( * )$ is fulfilled.
Definitions: 2 3
Examples: 4 5
Proofs: 7 8