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there's something confusing me about how do we divide two equation by each other? but may one is zero .. no?!

lets assume I have like this: (1) x=y (2)z=m how do we do (x/z)=(y/m) without caring about if z is zero or m is zero .. then we can't do division .. so how we do division ?! thanks a lot.

if z or m is zero then we can't do division .. so how we say that we can division two equation one by another?

quasi
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Tony.M
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    There is important context that you have failed to include in the question. In comments on the answers you indicate that your teacher performed a division and you are unsure how it was justified. This should be stated in the question itself. But it also matters exactly what division the teacher performed and where the things in the formulas came from. If the way you got $z$ and $m$ makes it impossible for them to be zero, you can divide without worrying about the “zero” case. – David K Aug 05 '19 at 20:38
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    Why haven't you asked the teacher as he was writing it? – kingW3 Aug 05 '19 at 20:47
  • @quasi, yes, I should open a new question for this. I included this in a comment because I thought it is relevant to the question. – NoChance Aug 05 '19 at 21:00
  • This discussion may be related to your question: https://math.stackexchange.com/questions/67994/why-should-you-never-divide-both-sides-by-a-variable-when-solving-an-equation?noredirect=1&lq=1 – NoChance Aug 05 '19 at 21:09

2 Answers2

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As you state, if $z = m = 0$ the division makes no sense. You'd have to add that restriction and handle the case $m = 0$ separately.

Without further details, we can't say anything more.

vonbrand
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  • You mean, once the teacher writes immediately we do division between two equations .. then he implicitly assume that z=m != 0 ? – Tony.M Aug 05 '19 at 19:35
  • @Tony.M He either assumes that $z=m\ne 0$; or he knows that this is the case from the context; or he restricts to the case that $m\ne 0$ and intends to treat the remaining case separately; or he is interested only in the (perhaps generic) case that $m\ne 0$. At any rate, the reason for this decision should be clarified (and you'd get points deducted if you committed the same in a test) – Hagen von Eitzen Aug 05 '19 at 19:59
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You certainly need to ensure that $z,m \neq 0$ if you are going to divide the equations. Often what is done is to assume $z=m=0$ and see if there is a solution. Whether you find one or not, you now assume that $z=m \neq 0$ and go ahead and divide because you have taken care of the $0$ case already.

Ross Millikan
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    But we do division immediately without saying z=m != 0 , I mean the teacher is solving that immediately without telling z=m != 0 .. so by sense we assume that z =m != 0?! – Tony.M Aug 05 '19 at 19:34
  • Yes, if you do the division you are assuming $z,m \neq 0$. One should account for that one way or another, but sometimes it gets overlooked. – Ross Millikan Aug 05 '19 at 19:37
  • So if its over looked then that's not true ... ?! but even if one side is zero and I do division .. where's the problem? 5 / 0 is infinity .. so we are fine .. no?! – Tony.M Aug 05 '19 at 19:41
  • No, division by zero is not allowed. It is not true that $5/0$ is infinity. You can say that informally when you really mean as a limit, but it is a good way to make mistakes if you are not careful. In your original question you can get into trouble because you get meaningless=meaningless, then can "cancel the 0s in the denominator" and introduce stray factors. – Ross Millikan Aug 05 '19 at 20:02
  • So why the teacher/other is overlook that issue that z=m =0 ?! I mean they simply say we do division between equations ...without telling the case of z=m=0 ! so we can neglect that?! – Tony.M Aug 05 '19 at 20:04
  • I can't say why people do things. Why do people drop signs when doing algebra? – Ross Millikan Aug 05 '19 at 20:22
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    @Tony.M Of all the people reading this, you are the only one who was sitting in the classroom and had a chance to see what the teacher did. And you are telling us almost nothing about that. So we cannot even say the teacher was right or wrong. – David K Aug 05 '19 at 20:42