Linear system/Elimination lemma/Fact/Proof

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

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

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

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

Then the equation

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

has the form

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

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