Algebra is the branch of mathematics that uses variables, constants, and operations to represent and solve relationships between quantities. It forms the foundation for higher mathematics, from calculus to linear algebra, and is essential in science, engineering, economics, and data analysis. At its core, algebra teaches you to work with abstract symbols instead of specific numbers, allowing you to solve problems that would be tedious or impossible with arithmetic alone. The key mental model: an equation is a balance β whatever you do to one side, you must do to the other β and mastering this principle unlocks virtually every algebraic technique you'll encounter.
What This Cheat Sheet Covers
This topic spans 24 focused tables and 203 indexed concepts. Below is a complete table-by-table outline of this topic, spanning foundational concepts through advanced details.
A jump-to index of every table row in this cheat sheet.
An interactive map of every table and concept in this topic.
Table 1: Fundamental Algebraic Expressions
The vocabulary everything else depends on. Before you can solve anything, you need to tell a variable from a constant, a term from an expression, and an expression from an equation β and know what makes terms "like" so you can combine them. The polynomial entries at the end set up the language used throughout the rest of the sheet.
| Concept | Example | Description |
|---|---|---|
x, y, n | A symbol representing an unknown or changing value β the cornerstone of algebra. | |
3x + 2y - 5 | β’ A combination of terms using operations β’ represents a value but is not an equation (no equals sign). | |
2x + 3 = 7 | β’ A statement that two expressions are equal β’ contains = and can be solved. | |
3x, -5yΒ², 7 | β’ A single mathematical unit separated by + or ββ’ can be a variable, constant, or product. | |
3 in 3x | β’ The numerical factor multiplying a variable β’ indicates how many times the variable is counted. | |
5, Ο, -3 | A fixed value that does not change. |