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This is the problem from my book I am working on.

Find a value for $A$, $B$ for which $Ax+By+C=0$ is not a line.

I got $0$ and $ \infty $, are there any others?

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    In my opinion, $A^2+B^2 \neq 0$ is the only condition.2012-08-29
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    @Siminore could you give me a reason why? Or at least point me in the correct direction2012-08-29
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    What is the book's definition of "a line"? $A^2+B^2=0\implies A=0$ and $B=0$, so if we have $C\ne0$ the equation is inconsistent, which is probably what @Siminore was thinking. Considering $\infty$ is not really appropriate here, I don't think many books would call $\infty$ a "value" (this terminology, and the fact that this is tagged as algebra-precalculus, suggests that "value" means "real number").2012-08-29
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    @MichaelBoratko its for A-levels so complex numbers should be allowed. The book just lists some various equations of lines and then has some problems.2012-08-29
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    @MaoYiyi If complex numbers are allowed then are $x,y\in\Bbb C$? If so, then many values don't represent lines in $\Bbb C$...2012-08-29
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    @MichaelBoratko I see what you mean.2012-08-29
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    The equation $ax+by+c=0$ is a line when both $a$ and $b$ are non-zero.2012-08-29
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    @Siminore: It isn't necessary that both $a$ and $b$ be non-zero. If $a=0$ and $b \neq 0$, then the line is horizontal; the other way around, it is vertical.2012-08-29
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    @Théophile Yes, I wrote a stupid thing. The correct version is "when *either* $a$ *or* $b$ (or both) are non-zero".2012-08-29

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If $A \neq 0$ or $B \neq 0$ then the equation is that of a line.

Otherwise, if $A = B = 0$, then there are two cases depending on the value of $C$. If $C=0$, then the equation becomes: $$0=0$$ which is true for all $(x,y) \in \mathbb R^2$; in other words, it describes the $x,y$-plane and not a line. On the other hand, if $C\neq 0$, then the equation is equivalent to $$0=1$$ which is not true for any $(x,y) \in \mathbb R^2$. The empty set is not a line.