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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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    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.2011-11-23

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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 $y$ou are not part of the solution you are part of the precipitate, as the old saying goes.2011-11-24
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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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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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The term that the textbook that we use for Algebra 1 is "contradiction" and the term used for an equation that has infinite solutions (4x + 2 =4x + 2) is called "identity"