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The problem statement lies, there is not more than one value of a for which the system has no solution (and even to get one, we have to assume b cannot be zero since otherwise there is always a trivial solution). It seems like bad question design to have written a system that admits the trivial solution and not mention that it is disallowed, since one could have just provided a system that actually doesn't allow the trivial solution by putting a non-zero on the RHS.
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Linear Algebra Foundations #8 - Systems of Equations
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The problem statement lies, there is not more than one value of a for which the system has no solution (and even to get one, we have to assume b cannot be zero since otherwise there is always a trivial solution). It seems like bad question design to have written a system that admits the trivial solution and not mention that it is disallowed, since one could have just provided a system that actually doesn't allow the trivial solution by putting a non-zero on the RHS.