Substitute
Pre-AlgebraTo 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 number in its place. Once you substitute, you can calculate the answer.
Example
If $n = 6$, substitute into $2n - 4$: write $2(6) - 4 = 12 - 4 = 8$.
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.
Definition
Substitution is the process of replacing a variable with a specific value or another expression. It is used to evaluate expressions, verify solutions, and is the basis for the substitution method in solving systems of equations.
Example
To check whether $x = 3$ solves $2x + 1 = 7$: substitute $x = 3$ to get $2(3) + 1 = 7$. True, so $x = 3$ is a solution.
Key Insight
When you substitute one expression for another (not just a number), you are doing algebraic substitution. For example, substituting $u = x^2$ into a complicated expression can make it simpler to work with.
Definition
Algebraic substitution is the application of a variable map (homomorphism) to an expression, replacing free occurrences of a variable with a term. In formal logic, simultaneous substitution of all free variables by terms produces a ground term. In calculus, substitution (u-substitution) is a change of variable that transforms one integral into an equivalent, easier one via the chain rule in reverse.
Example
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 essentially a relabeling. Its power comes from the fact that equivalent expressions remain equivalent after any valid variable substitution, which is the basis of formal algebraic manipulation.