Linear system/Elimination lemma/Fact/Proof

Proof

Changing the numbering, we may assume ${}x=x_{1}$. Let ${}G$ be the equation

${}ax_{1}+\sum _{i=2}^{n}a_{i}x_{i}=b\,$

(with ${}a\neq 0$), and ${}H$ be the equation

${}cx_{1}+\sum _{i=2}^{n}c_{i}x_{i}=d\,.$

Then the equation

${}H'=H-{\frac {c}{a}}G\,$

has the form

${}\sum _{i=2}^{n}{\left(c_{i}-{\frac {c}{a}}a_{i}\right)}x_{i}=d-{\frac {c}{a}}b\,,$

and ${}x_{1}$ does not occur in it. Because of ${}H=H'+{\frac {c}{a}}G$, the systems are equivalent.

To fact