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Given a system of linear equations, is it true that if the associated homogeneous system has unique solution (the trivial one), then the system itself has a unique solution?

If this is true, it would mean that there is a mistake in the table from this Wikipedia page (which seems to be extracted from Jim Hefferson's Linear Algebra book), so I think this is not true, but how to find a counter-example?

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No, it is not true. Consider, for example, the following set of equations $$x+y=2,\tag 1$$ $$x-y=0, \tag {2}$$ and $$x+2y = 2. \tag 3$$

Obviously, the homogeneous solution is unique, i.e. $x=y=0$. But the solution of the linear system does not exist.

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