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In the 10th ed. of Elementary Linear Algebra (Anton), the following statement exists:

If Ax = b is a system of linear equations, exactly one of the following is true: (a) the system has no solutions, (b) the system has exactly one solution, (c) the system has more than one solution. The proof will be complete if we can show that the system has infinitely many solutions in case (c)

Why is the text in bold true? How does showing that (c) has infinitely many solutions prove that the system of linear equations may have no solutions, or only one solution?

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    $\begingroup$ This looks like part of a proof of a statement, but you did not copy the statement in the question. Please include more context; currently the question does not really make sense. $\endgroup$ Apr 29, 2014 at 12:36

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Conditions $a,b,c$ are mutually exclusive and complete. The author is about to prove that $c$ implies that there are infinitely many solutions. Therefore the conditions $a,b,c'$ are mutually exclusive and complete, where $c'$ is that there are infinitely many solutions.

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"The proof" does not refer to the proof of the statement that a system has zero solutions, one solution, or more than one solution - this statement is trivial. I don't have the book, so I don't know what statement the bold sentence is completing the proof of.

My guess would be that the author is proving that a system has either no solutions, one solution or infinitely many solutions. It is not necessary for that statement to prove that each of these situations actually occurs (although they do), merely that the number of solutions cannot be finite and greater than one; this is exactly the statement in bold.

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