Algebra

Overview

Algebra is the branch of mathematics that deals with abstract systems, i.e. algebraic structures, and the manipulation of expressions within those systems. It is a generalization of arithmetic that introduces variables and other algebraic operations.

Equations

An equation is a relation between two expressions that may be true for some values of the variables. To solve an equation means to find all of the values for the variables that make the two expressions on either side of the equation equal to one another.

A functional equation is an equation that expresses a relationship between the values of a function at two or more points.

An implicit equation is a relation of the form $F(x_1,\dots ,x_n)=0$, where $F$ is a function of several variables. For example, the implicit equation of the unit circle is $x^2+y^2-1=0$.

An identity is an equation that is true for all allowable values of the variables.

Bibliography

• Ted Sundstrom and Steven Schlicker, Trigonometry, 2024.
• Tom M. Apostol, Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra, 2nd ed. (New York: Wiley, 1980).
• Wikipedia. “Algebra.” July 1, 2025. https://en.wikipedia.org/w/index.php?title=Algebra.
• Wikipedia. “Implicit function.” November 30, 2025. https://en.wikipedia.org/w/index.php?title=Implicit_function.