solv

The root solv comes from Latin solvere, "to loosen, untie, release." The physical picture is concrete: a knot, a rope, a bond — and the act of un-doing it. Latin formed solvere itself from se- (apart) + luere (to loosen), so even at birth the word meant "to loosen apart." Its past participle was solūtum, which is why the family splits into two spellings: the verb stem solv- and the participle stem solu-/solut-.

Once you hold the image of "loosening a knot," the whole family lines up:

- solve = loosen a knot in your mind. A hard problem is a tangle; to solve it is to undo the tangle. A solution is either the untangled answer, or — in chemistry — the way a solid "loosens apart" and disperses into a liquid. Same word, two worlds: the answer to a problem and salt vanishing into water.
- dis- + solvere → dissolve: loosen something completely apart until it falls to pieces — sugar dissolving in tea, or a parliament dissolving into nothing. dissolution is the noun.
- re- + solvere → resolve: literally "loosen again." Two senses grew from this. One stayed close to solve: to resolve a dispute is to untangle it. The other jumped: when you "loosen" all your hesitations and settle your mind into one firm position, you resolve to act — hence resolution (a firm decision, a formal vote) and resolute (unwavering). The same loosening that frees a knot also frees you from doubt.
- ab- + solvere → absolve: loosen someone away from (ab-) their guilt or obligation — to forgive. absolution is the priest's formal release from sin.
- absolute is the family's biggest leap. Latin absolūtus meant "loosened away from everything" — unattached, unconditioned, depending on nothing. From "free of all ties" came the modern sense: complete, total, unlimited. Absolute power answers to nothing; absolute zero is bound by nothing further. The original "loosening" is almost invisible now, but it is exactly why "absolute" means not relative.
- solvent / insolvent carry the loosening into finance. To be solvent originally meant being able to loosen (pay off, discharge) your debts; an insolvent company cannot untie its obligations — it is bankrupt. The chemistry sense of solvent (a liquid that dissolves things) and the financial sense are the same metaphor: a solvent dissolves a solid; a solvent person dissolves a debt.

The constant across the whole family is the act of loosening: untangle a problem (solve), break a solid apart in liquid (dissolve), settle your mind by loosening doubt (resolve), free someone from guilt (absolve), free from all conditions (absolute), discharge a debt (solvent).