Substituting

Substituting is the act of replacing one element with another. In mathematics and related fields, substitution refers to replacing a variable, symbol, or expression with a substitute that preserves the problem’s structure or meaning. The goal is to simplify calculations, reveal relationships, or adapt models to data or requirements.

In mathematics, substitution has several key uses. In algebra, the substitution method solves systems of linear equations by expressing one variable in terms of another and substituting into the other equation. In calculus, u-substitution (integration by substitution) rewrites an integral in terms of a new variable to simplify integration, with careful handling of differential changes. Substitution also underpins many algebraic manipulations, where equivalent expressions are obtained by replacing parts of an expression with alternatives that yield the same value.

A simple example: solve x + y = 5 and 2x − y = 1. From the first equation, x = 5 − y. Substituting into the second gives 2(5 − y) − y = 1, which leads to y = 3 and then x = 2.

Beyond mathematics, substitution is used in logic and computer science to replace variables with terms within formal expressions, as in substitutions that enable term rewriting and unification. In chemistry, substitution refers to reactions where an atom or group in a molecule is replaced by another. In linguistics, substitution can describe the replacement of sounds or lexical items in language data or analysis. Across disciplines, substitution serves as a fundamental technique for transforming problems into more tractable form.