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I was wondering if there is a formal name for the equations which don't have any solution?

For example consider this equation in $m$ :

$$ -2(3-m)+15=6m-4(m-20)$$

If we do the algebra we will get $2m-6m+4m=80-15+6 \Rightarrow 0=71$ which implies no solution of $m$.

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Such an equation is called "inconsistent" –  Bill Cook Nov 23 '11 at 19:18
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Surely the equation is insoluble, and to be inconsistent it would have to be a set of equations? –  Peter Taylor Nov 23 '11 at 19:21
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I always thought the formal name was "equations with no solutions". –  Arturo Magidin Nov 23 '11 at 19:24
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I see no problem using the term "inconsistent" for a single equation in the same way that a single linear equation is still a "system" of equations. –  Bill Cook Nov 23 '11 at 19:24
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Also, the term "inconsistent" does apply quite well. If the equation reduces to "0=1" then it is not consistent with itself. "Unsolvable" is ok. But this can conjure up notions of unsolvability in the sense of Galois theories etc. –  Bill Cook Nov 23 '11 at 19:30
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4 Answers 4

up vote 6 down vote accepted

It seems that unsolvable, insolvable and insoluble are all used in this sense. Personally I'd prefer either of the first two, to avoid confusion with the alternative meaning of insoluble in physics and chemistry.

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If you are not part of the solution you are part of the precipitate, as the old saying goes. –  drxzcl Nov 24 '11 at 8:28
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A set of equations with no solutions is called inconsistent if there is no simultaneous solution for the set.

It is important to note that a set containing one element is still a set, i.e. $ 0 = 71 $ is shorthand for $\{ 0 = 71 \}$ (a notation which is avoided due to obvious reasons involving tediousness of writing) and this set of equations is inconsistent.

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Others have noted inconsistent (adj.), which may be best. Perhaps as good is unsatisfiable (adj.). False (adj.) and a falsehood (n.) might work, too. In basic logic, a sentence always false (like $p\wedge\neg p$) is called a contradiction (n.), which I suppose can be used here, too.

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According to p. 185 of Basic Math: A Combined Version by Williams, Miller, Salzman, and Lial (HarperCollinsCustomBooks, 1992), an equation that is a false statement for every value of the variable is an inconsistent equation or a contradiction.

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