If $a = \frac{x}{b}$ and $a = \frac{c}{b}$, and I solve for $x$ I get $x = c$.

$b$ has been removed because it appeared in the numerator and the denominator.

What is it called in English what happened to $b$?

$b$ ...

In German I would say "$b$ kürzt sich weg."

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  • Simpler: $\color{#c00}{b^{-1}} x = \color{#c00}{b^{-1}} c\,\Rightarrow\, x=c\,$ by cancelling $\,\color{#c00}{b^{-1}},$ i.e. we can apply the Cancellation Law (rather than repeating its proof inline) – Number Sep 14 at 15:35
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    Note that your solution is incomplete. You are still left with the condition $ b \ne 0$ that must be tacked onto every conclusion that follows. – jpmc26 Sep 14 at 17:23
  • @jpmc26 Denominators of fractions are nonzero by definition. – Number Sep 14 at 17:54
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    @Number 1. Once you cancel it, it's not obvious there is a denominator. Stating the restriction explicitly makes it more clear that the condition applies to everything that follows. 2. $b$ may not have been defined as a denominator. The restriction still applies to conclusions that follow. 3. As this is an educator site, it's important to note that students can easily miss this important detail. 4. Depending on the starting equations, $b = 0$ may yield some other solution that needs to be found. Keeping in mind that this one doesn't work for that case helps you remember to look for it. – jpmc26 Sep 14 at 18:01
  • liguistically I understand cancel and eliminate as same thing. Doesn't even give me pause. ceemrr.com/Geometry2/Eliminating_Fractions/… (many other examples available...GIYF) – guest Sep 14 at 18:11
up vote 11 down vote accepted

If you continue the operations until $$\require{cancel}\frac{x}{b}=\frac{c}{b},\qquad\left(\frac{x}{b}\right)b=\left(\frac{c}{b}\right)b,\qquad x\left(\frac{b}{b}\right)=c\left(\frac{b}{b}\right),\qquad x\left(\frac{\cancel b}{\cancel b}\right)=c\left(\frac{\cancel b}{\cancel b}\right)$$ then $x=c$, I would say that you cancelled the common factor $b$ in the fraction.

  • I would also say cancelled, but some folks prefer "reduced" as to distinguish from additive cancellation. – James S. Cook Sep 16 at 18:50

So German "$b$ kürzt sich weg" becomes in English "$b$ cancels out". We may also say "$b$ is eliminated".

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    I've never heard "eliminated" used as a synonym for "cancelled". Do you have any links exhibiting such? – Number Sep 14 at 15:45
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    @Number I have heard it, but only from professors who are not native English speakers and are translating from their native languages. (That's not to say that it's incorrect, I just found it to be less common.) – Moshe Katz Sep 14 at 17:30
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    I am used to either cancel or eliminate and it does not sound odd to my ear and I am a native English speaker. ceemrr.com/Geometry2/Eliminating_Fractions/… – guest Sep 14 at 18:11
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    See for example en.wikipedia.org/wiki/… It is true that "cancel" and "eliminate" sometimes have different meanings, but they both qualify as "variables disappear when solving". – Gerald Edgar Sep 14 at 18:14
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    @guest e.g. we can cancel $a$ from $\dfrac{ac+a^2}a$ to get $c+a$ but "eliminate $a$" is not what we did. – Number Sep 14 at 22:34

In general, as others have noted, if you have an equation such as $$\frac{x}{b}=\frac{c}b$$ The step to get from there to $$x=c$$ is typically referred to as cancelling the denominator. More generally, if you can just remove some piece of the equation, you can use the verb "cancel" both with that piece as the object and the subject. For instance:

  • We cancel the denominators.

  • The denominators cancel.

You can also use the phrasal verb "cancel out" as in "The denominators cancel out."

  • I've never heard that language used. Do you have links to its use in respectable texts? it is incorrect terminology because it is not the denominator $b$ that is being cancelled but instead its inverse $b^{-1} = 1/b$ Usually cancelling the denominator refers to $\,bc/b = c,$ i.e. cancelling $b$ from $bx = bc$ $\quad$ – Number Sep 14 at 17:06
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    @Number Are you sure you've never heard this? It sounds like you're trying to be pedantically "correct" but not correct in terms of actual usage. "The b's cancel [out]" is surely a very common way that someone would describe this scenario. Certainly how I would say it. Usage is what matters, not technicality. And I'm unsure why "respectable texts" matter. Most people don't speak like how a textbook reads... – only_pro Sep 14 at 19:15
  • @only_pro I refer specifically to "cancelling the denominator(s)". For example, precisely what does the mean when applied to $\dfrac{bx}b = \dfrac{bc}b$? Does it yield $bx = bc$ (as in this answer) or $x = c$? – Number Sep 14 at 19:19

Coming at this from a slightly more abstract point of view than the other answers, this is an application of the multiplicative cancellation law (over the rationals if $a$, $b$, and $c$ are all integers; or over the reals if $a$, $b$ and $c$ are real numbers; or over the complex numbers; or whatever...). Specifically, in this context, the cancellation law says:

Let $q$, $r$, and $s$ be rational numbers (or real numbers, or complex numbers) with $q\ne 0$. Then $q\cdot r = q\cdot s$ if and only if $r=s$.

Note that it is quite important here that $q \ne 0$. If $q = 0$, then both $r$ and $s$ may be chosen freely, and no cancellation is possible. Taking $q = \frac{1}{b}$ (assuming that $b\ne 0$, $q$ is well-defined and we automatically have $q\ne 0$), $r = x$, and $s = c$, the multiplicative cancellation law gives us $$ \frac{x}{b} = \frac{c}{b} \iff \frac{1}{b} \cdot x = \frac{1}{b} \cdot c \iff x = c. $$ Because we are using the multiplicative cancellation law, the process is called cancelling or cancelling out the common factor (in this case, we are cancelling a factor of $\frac{1}{b}$). Indeed, I think that a properly rigorous reading of this step would be "We cancel out a common factor of $\frac{1}{b}$."

It might also be reasonable to say that "We cancel the common factors from the denominators," or more simply "We cancel the denominators." That said, because I am kind of pedantic, I would be a little hesitant to say that anything is being done to $b$. We aren't really cancelling a factor of $b$, but rather a factor of $\frac{1}{b}$.

  • +1 Thanks for elaborating on the point I emphasized in comments – Number Sep 14 at 18:21

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