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I'm a high school maths teacher.

I was wondering if there is a name for all equations where a variable appears only once e.g.

$$\frac{(x+3)^2}{4}=5\qquad {\rm{or}} \qquad \frac{1}{2}+x=\frac{3}{4}$$

I was planning on introducing the "onion method" for solving to my students and lots of them like knowing the names of equations.

Informally, I would call these arithmetic equations since arithmetic is all that's required to solve them.

Thanks, Ben

Ben Crossley
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  • If $x$ is in one side, you can isolate it. Not sure if it still applies for $x + 3 = 1$. – soupless Dec 09 '21 at 09:06
  • @soupless $x+3=1$ is still an equation with $x$ appearing only once. This equation would be in the same class as the two I provided. – Ben Crossley Dec 09 '21 at 09:21
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    Does "single variable equation" work? technically it's not, since $x^2+x=0$ is also an equation in a single variable. Also, there's an SE for Mathematics educators in case it interests you – Adil Mohammed Dec 09 '21 at 09:28
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    I wouldn't call them arithmetic but algebraic because "arithmetic" would mean that coefficients are necessarily integers of ratio of integers. I like the "onion" expression. The idea is that you consider as valid any composition of translations $x \to x+a$, dilations $x \to bx$, $x \to x^n$, elevation to a power, $x \to \sqrt[n]{x}$ in any order... – Jean Marie Dec 09 '21 at 09:31
  • @AdilMohammed $x^2 + x = 0$ does not work since $x$ 'appears' more than once. By which I mean, there are quite literally 2 $x$s in that equation. – Ben Crossley Dec 09 '21 at 09:31
  • @JeanMarie Almost. You could happily have an equation such as $\sqrt{3}x + \sqrt{2} = 5$ – Ben Crossley Dec 09 '21 at 09:32
  • I wasn't restricting $a,b$ etc. to the domain of integers... – Jean Marie Dec 09 '21 at 09:36
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    I think you are probably rather looking for a name of the form and not the equation, since the equations $\frac{(x+3)^2}{4}=5$ and $\frac{(x+3)(x+3)}{4}=5$ are equivalent. So, maybe saying those equations are in "onion form" would be a candidate? – Andreas Lenz Dec 09 '21 at 09:42
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    @JeanMarie Maybe I have misunderstood what you meant by coefficients are "necessarily integers or ratio of integers". I took that to mean coefficients could not be surds. I was also steering clear of algebraic equations since they are already a well-established class of equations. – Ben Crossley Dec 09 '21 at 09:42
  • @AndreasLenz More than happy to call it onion form. Students would likely remember it too. – Ben Crossley Dec 09 '21 at 09:43
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    @BenCrossley You can use the tag [terminology] for this question. Not sure what other tags you should use, but [education] works. – soupless Dec 09 '21 at 10:03

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