Linear system/Elimination lemma/Fact

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Elimination lemma

Let denote a field, and let denote an (inhomogeneous) system of linear equations in the variables . Suppose that is a variable which occurs in at least one equation with a coefficient .

Then every equation , different from ,

can be replaced by an equation in which does not occur any more, and such that the new system of equations , which consists of and the equations , is equivalent

with the system .