Is there a formal name for an equation that has no solution?
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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$\begingroup$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.
$\endgroup$ 1 $\begingroup$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.
$\endgroup$ $\begingroup$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.
$\endgroup$ $\begingroup$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.
$\endgroup$ $\begingroup$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"
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