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A recent question to when to use of and when for/to for solution suggested that of appears only in context of chemistry, and the word means something very different then. But I recalled almost universally finding

−2 and 2 are the solutions of the equation x² = 4.

I consulted the dictionary and to my surprise I found:

2. Mathematics A value or values which, when substituted for a variable in an equation, make the equation true. For example, the solutions to the equation x² = 4 are 2 and −2.

I consulted the ngram and it made me even more confused than before.

ngram

It seems like solution to the equation is a relatively new trend, but picking up.

So, what’s the official stance on correctness of the preposition to go with solution in mathematics?

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    Prepositions are versitile words with overlapping meanings. There are several usages in English where more than one preposition could be used, with little or no change in meaning. Let me reiterate that: There are several usages in English where more than one preposition could be used, with little or no change of meaning. The Ngram is interesting, but hardly surprising. There's no "official" correct preposition here.
    – J.R.
    Commented Oct 22, 2012 at 7:11
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    @J.R. I think your comment is an answer in and of itself.
    – Jim
    Commented Oct 22, 2012 at 8:12
  • Google nGrams need careful inference, especially if the parameter is not sharply and strongly defined. Usage of solution: a. an object that is the solution b. the process of solving -- accordingly, the context determines the preposition. This is just one of the factors influencing the choice of the preposition; similarly, there are others as well.
    – Kris
    Commented Oct 22, 2012 at 8:25
  • Comments should help OP and editors to improve the question. Answers should be posted as answers.
    – MetaEd
    Commented Oct 22, 2012 at 12:18
  • Please add a live link to the ngrams you posted. I have been unable to replicate what you show. Ngrams for solution of the equation,solution to the equation,solution for the equation is different, particularly 1930-1940
    – James Waldby - jwpat7
    Commented Oct 22, 2012 at 15:02

2 Answers 2

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Prepositions are versatile words with overlapping meanings. There are several usages in English where more than one preposition could be used, with little or no change in meaning. Let me reiterate that: there are several usages in English where more than one preposition could be used, with little or no change of meaning.

The Ngram is interesting, but hardly surprising. There's no "official" correct preposition here.

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I think there is a slight difference in meaning: solution of indicates that some set of solutions is known to exist, while solution to reflects no such knowledge. Consider these examples:

A. I have not found any solutions of this equation.

B. I have not found any solutions to this equation.

My intuition is that in (A), the speaker is treating it as a given that the equation has solutions, and just can't find them; in (B), however, it is possible that there are no solutions.

This intuition maybe comes from the fact that only mathematical systems can be said to definitively possess solutions. Solution to is much more common than solution of when the word is used in the context of real-world issues: a solution to the debt crisis.

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  • Interesting, but I'm not sure that most English speakers would recognize such a distinction.
    – Jay
    Commented Oct 22, 2012 at 14:03
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    I think almost no English speakers would recognize such a distinction. Neither A nor B suggest to me that the speaker assumes the equation has solutions. Both suggest the possibility of there being no solutions.
    – James Waldby - jwpat7
    Commented Oct 22, 2012 at 18:35

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