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In Why do we believe the equation ax+by+c=0 represents a line?, I wrote this answer:

Let $A=\left(x_{A},y_{A}\right)$ and $B=\left(x_{B},y_{B}\right)$ be distinct points.

We define the line $l$ through points $A$ and $B$ to be the set of points $P=\left(x,y\right)$ such that $\overrightarrow{BP}\parallel\overrightarrow{AB}$ (i.e. $\left(x-x_{B},y-y_{B}\right)=k\left(x_{B}-x_{A},y_{B}-y_{A}\right)$ for some $k\in \mathbb R$).

With some algebra, we can show $l=\left\{ \left(x,y\right):ax+by+c=0\right\}$ (for some $a,b,c\in\mathbb{R}$ with at least one of $a$ or $b$ non-zero).

I think my answer is helpful and does not contain any errors. But it seems I am mistaken.

Could those who deleted my answer tell me what's wrong with my answer so I can write better answers in the future? Thanks!

(I wonder also if instead of simply deleting my answer without comment, one could have posted some comments/explanations on why my answer is bad and perhaps given me an opportunity to improve my answer.)

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    $\begingroup$ Note that the Question you responded to is more than 10 years old and has more than one upvoted Answer from back around that time. Recently new Answers were posted, and this causes the Question to get bumped back to the front page. In any case I think you were somewhat unlucky in that more than one recent Answer was of poor quality and needed to be deleted, and your Answer (because it did not clearly highlight what new information was added or how it resolves the stated problem) got somewhat swept into the mix. I did not vote to delete, so take my points as speculative as to other's votes. $\endgroup$
    – hardmath
    Feb 26 at 6:25
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    $\begingroup$ I wouldn't have recommended deletion, but I would have downvoted and asked you not to answer old questions with good answers unless you had a lot to add, since (as noted) your answer bumps the question to the active queue. $\endgroup$ Feb 28 at 19:07
  • $\begingroup$ See here $\endgroup$
    – Shaun
    Mar 3 at 14:35
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    $\begingroup$ The answer has now been undeleted, so you have an opportunity to revise it if you wish. $\endgroup$
    – hardmath
    Mar 4 at 1:21
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    $\begingroup$ In this aspect, I am curious, what is wrong with bumping old questions with an answer that you think adds something to the Q&A? $\endgroup$
    – Max0815
    Mar 10 at 18:59

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