# Solution of/to/for equation

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. 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?

• 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.
Oct 22, 2012 at 7:11
• @J.R. I think your comment is an answer in and of itself.
– Jim
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
Oct 22, 2012 at 8:25
• Comments should help OP and editors to improve the question. Answers should be posted as answers. 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 Oct 22, 2012 at 15:02

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.

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: