Two jobs, one picture
In ordinary English a solution is either a way of settling a problem or a mixture in which one substance is uniformly dispersed in another, most often a liquid. Mathematics keeps a narrower version of the first job: the values that satisfy an equation. Chemistry keeps the second in three related uses, the process of dissolving, the resulting mixture, and the state of being dissolved. Business English later stretched the problem sense into a label for a product or service. The history of the noun shows those jobs are two results of the same older image: something bound is set free.
Borrowed, not inherited
The origin of the word "Solution" is a late medieval borrowing. Native English already had verbs of loosening and losing from Germanic stock, but this noun arrived from the Continent. In the late fourteenth century scribes wrote solucioun, solucion, and similar spellings. Those forms match Old French solucion, the ancestor of modern French solution, and Latin solutio, whose accusative solutiōnem is the shape that usually feeds Romance and English nouns in -tion.
Major historical dictionaries treat the immediate source as genuinely double. The noun may have come through Anglo-French, or it may have been taken straight from Latin, or English may have been drawing on both at once. First English evidence belongs to the Middle English decades around the 1380s. One early witness is the Wycliffite Bible, where solucioun names the telling of a royal dream. Scottish and English writers of the same generation already use the word for the working-out of a difficulty. The spelling solution, with t rather than c, becomes standard in the sixteenth century.
Old French had already gathered several Latin senses under one heading: explanation, the breaking up or dissolving of matter, and payment. English inherited that spread almost immediately rather than inventing a single tidy meaning and then branching later.
A Latin compound of coming apart
The Latin noun is an action name built on the past-participle stem of solvō, solvere: to loosen, untie, undo, or dissolve. Morphologically it is solūt- plus the suffix -tiō, the source of English -tion, which turns a verb into the act or its result. The verb itself is a compound. Latin se- marks motion apart or away; luō is to unbind or set free. Behind luō stands a Proto-Indo-European root of loosening, dividing, or cutting apart, written in different handbooks as *leu- or *lewh₁-. Some etymologies present a reconstructed *se-lu- from the reflexive stem *s(w)e- plus that root. The reconstructions differ in packaging; they do not disagree about what Roman speakers were doing with the verb.
Solvere was a busy word. It could unfasten a knot, open a letter, slacken a bond, dismiss troops, weigh anchor, explain a question, or pay a debt. That last use is not a metaphor piled on later. To pay was to loosen an obligation. English relatives from the same Latin family keep different slices of the range: solve, solvent, solute, soluble, dissolve, absolute, resolute, insolvent. Chemistry took the physical mixing. Law and money kept the idea of discharging what is owed. Problem-solving kept the untying of a knot.
Cognates and cousins make the same family visible outside English. French solution, Spanish solución, and Italian soluzione are the expected Romance siblings. At a deeper level the old loosening root also feeds Greek lyein, to unbind, and lysis, a loosening; Sanskrit lunati, to cut off; and Germanic words for what is loose, including English loose and lose. German Lösung and Dutch oplossing are not borrowings of the Latin noun. They are independent builds from the Germanic verb of loosening, and they happen to carry the same double load of answer and liquid mixture.
How the senses divided
Once the noun was in English, trades pulled the loosening image in different directions. The explanation sense stayed closest to everyday talk: an answer to a question, a method for a difficulty, later the working of a mathematical problem. Dream interpretation, prominent in the earliest biblical uses, faded as a special application rather than a separate word.
A legal and fiscal sense, solution as payment or the satisfaction of a claim, is well recorded from the late fifteenth century into older Scots and English law. It is now rare outside historical writing. Another lost thread treated solution as release or setting free, including deliverance from an oath and, in philosophical prose, the parting of soul from body. Medical writers sometimes used the noun for the crisis or ending of a disease.
Surgeons kept a set phrase, solution of continuity, for a wound or rupture that separated parts of the body that should have remained joined. English has it by the 1540s as a rendering of Galenic language. From medicine the phrase moved into figurative prose for any breach or interruption, from glass to narrative to public affairs. It is a reminder that dissolving and cutting were never far apart in the Latin verb.
The material sense is equally old in English, even if the laboratory mixture is the one modern speakers notice first. Around 1390 John Gower lists solucion among alchemical operations beside distillation and congelation: the breaking of a solid into a fluid state. That is the process. The countable liquid, a preparation of a dissolved substance in a solvent, is clearly attested by the 1590s. Later scientific English added the state of being dissolved, often in the collocation in solution. The homogeneous mixture of modern physical chemistry is a tightening of that early modern usage, not a new coinage.
Speakers continued to feel, even dimly, that solving a problem and dissolving a salt are both kinds of unbinding. The recent commercial habit of calling a software package a solution is only the problem sense wearing a sales jacket.
The German parallel
A useful check on the English story is the German noun Lösung. It answers both to a solved puzzle and to a chemical mixture, and it does so without borrowing Latin solutio. That parallel is evidence that the double sense is not an accident of French spelling. It is what a loosening-root tends to do when a culture talks about knots, debts, bodies, and salts with the same verb. English merely imported the Latin name for that work, then kept both of the jobs.