Substitute
To substitute in algebra means to replace a variable with a specific number or expression in order to evaluate or simplify.
Definition
To substitute means to swap out a variable and put a specific number, or another expression, in its place so you can calculate or verify an answer. Substitution is used to evaluate expressions, to verify solutions, and it is the basis of the substitution method for solving systems of equations. Formally, algebraic substitution is the application of a variable map (a homomorphism) to an expression, replacing free occurrences of a variable with a term; in formal logic, simultaneously substituting all free variables by terms produces a ground term, and in calculus, u-substitution changes variables to transform one integral into an easier, equivalent one via the chain rule in reverse.
Example
If $n = 6$, substitute into $2n - 4$ to get $2(6) - 4 = 12 - 4 = 8$. To check whether $x = 3$ solves $2x + 1 = 7$, substitute $x = 3$ to get $2(3) + 1 = 7$, which is true. For u-substitution, to evaluate $\int 2x\sqrt{x^2+1}\,dx$, let $u = x^2 + 1$, $du = 2x\,dx$; the integral becomes $\int \sqrt{u}\,du = \frac{2}{3}u^{3/2} + C = \frac{2}{3}(x^2+1)^{3/2} + C$.
Key Insight
Substitution is just like replacing a nickname with a person's real name: the sentence still makes sense, and now it refers to something specific. When you substitute one expression for another rather than just a number, for example letting $u = x^2$ in a complicated expression, you can make it far simpler to work with; this power comes from the fact that equivalent expressions remain equivalent after any valid variable substitution, the basis of formal algebraic manipulation.