Linear system/Elimination lemma/Fact/Proof

Proof

Changing the numbering, we may assume $x=x_1$. Let $G$ be the equation

$a{x}_1+\sum _{i=2}^n{a}_i{x}_i=b$

(with $a\ne 0$), and let $H$ be the equation

$c{x}_1+\sum _{i=2}^n{c}_i{x}_i=d.$

Then the equation

$H^{\prime }=H-\frac{c}{a}G$

has the form

$\sum _{i=2}^n(c_i-\frac{c}{a}{a}_i)x_i=d-\frac{c}{a}b,$

and $x_1$ does not occur in it. Because of $H=H^{\prime }+\frac{c}{a}G$, the systems are equivalent.

To fact