solve

Solve is a verb meaning to find an answer to a problem, explanation to a question, or the values that satisfy an equation; also used metaphorically to mean resolving a dilemma or dispute. The noun solution refers to the answer or to a homogeneous mixture in chemistry; the related adjective solvable describes problems that can be solved.

In mathematics, solving typically means determining all values that satisfy a given statement, equation, or system of equations. For a linear equation ax=b, the solution is x=b/a (if a ≠ 0). For systems Ax=b, a solution is a vector x satisfying the equation. Common methods include substitution, elimination, factoring, and matrix techniques such as Gaussian elimination or matrix inversion; numerical methods (Newton's method) can be used when exact solutions are not available.

In other domains, solving encompasses puzzle-solving, algorithmic problem solving, and engineering design. In computing, problem solving is often expressed as designing algorithms and proofs of correctness, with tools such as solvers, constraint programming, or optimization libraries.

Etymology: solve derives from Latin solvere, meaning to loosen or untie, via Old French solver, and entered English in the Middle Ages. The concept is tied to finding an answer, a method, or a resolution.