Linear system/Elimination lemma/Fact

Elimination lemma

Let ${}K$ denote a field, and let ${}S$ denote an (inhomogeneous) system of linear equations over ${}K$ in the variables ${}x_{1},\ldots ,x_{n}$. Suppose that ${}x$ is a variable which occurs in at least one equation ${}G$ with a coefficient ${}a\neq 0$.

Then every equation ${}H$, different from ${}G$,

can be replaced by an equation ${}H'$, in which ${}x$ does not occur any more, and such that the new system of equations ${}S'$ that consists of ${}G$ and the equations ${}H'$, is equivalent

with the system ${}S$.

Proof, Write another proof