Algebra Fundamentals

Algebra

Definition: A branch of mathematics dealing with symbols and the rules for manipulating those symbols. It includes solving equations and understanding mathematical relationships.
Key Concepts:
- Variables: Symbols (like x, y) used to represent unknown values.
- Constants: Fixed values (like numbers) that do not change.
- Expressions: Combinations of variables and constants using operations (e.g., 3x + 2).
- Equations: Mathematical statements asserting that two expressions are equal (e.g., 2x + 3 = 7).
Operations:
- Addition and Subtraction: Combining or removing values.
- Multiplication and Division: Scaling values and distributing them.
- Exponentiation: Raising a quantity to a power (e.g., x^2).
Types of Algebra:
- Elementary Algebra: Basic operations and expressions, solving linear equations.
- Abstract Algebra: Studies algebraic structures such as groups, rings, and fields.
- Linear Algebra: Concerns vector spaces and linear mappings between them.
Linear Equations:
- Form: ax + b = 0, where a and b are constants.
- Solutions: Found by isolating the variable (e.g., x = -b/a).
Quadratic Equations:
- Form: ax^2 + bx + c = 0, where a, b, and c are constants, and a ≠ 0.
- Solutions: Found using:
  - Factoring
  - Completing the square
  - Quadratic formula: x = (-b ± √(b² - 4ac)) / (2a)
Functions:
- Definition: A relation that assigns exactly one output for each input.
- Notation: f(x), where f is the function and x is the input.
- Types include linear functions (y = mx + b), polynomial functions, and exponential functions.
Factoring:
- Breaking down expressions into products of simpler expressions (e.g., x^2 - 1 = (x - 1)(x + 1)).
- Useful in solving equations and simplifying expressions.
Systems of Equations:
- A set of equations with multiple variables.
- Solutions can be found using:
  - Substitution method
  - Elimination method
  - Graphical method
Inequalities:
- Expressions that use inequality signs (<, >, ≤, ≥).
- Solutions represent ranges of values.
Applications:
- Used in various fields such as physics, engineering, economics, and more to model real-world problems and relationships.

Definition of Algebra

Branch of mathematics focused on symbols and rules for manipulation.
Involves solving equations and understanding relationships between mathematical elements.

Key Concepts

Variables: Symbols (e.g., x, y) representing unknown values.
Constants: Fixed values that remain unchanged (e.g., numbers).
Expressions: Combinations of variables and constants using mathematical operations (e.g., 3x + 2).
Equations: Mathematical assertions of equality between two expressions (e.g., 2x + 3 = 7).

Operations in Algebra

Addition and Subtraction: Combine or remove values.
Multiplication and Division: Scale values or distribute them.
Exponentiation: Raise a quantity to a specified power (e.g., x²).

Types of Algebra

Elementary Algebra: Focuses on basic operations, expressions, and solving linear equations.
Abstract Algebra: Explores algebraic structures like groups, rings, and fields.
Linear Algebra: Studies vector spaces and linear mappings.

Linear Equations

General Form: ax + b = 0, where a and b are constants.
Finding Solutions: Isolate the variable to determine values (e.g., x = -b/a).

Quadratic Equations

General Form: ax² + bx + c = 0, with a, b, and c as constants and a ≠ 0.
Solution Methods:
- Factoring: Break down into simpler components.
- Completing the Square: Rearrange to create a perfect square.
- Quadratic Formula: x = (-b ± √(b² - 4ac)) / (2a).

Functions

Definition: Relation assigning one output for each input.
Notation: f(x) represents the function with x as the input.
Types: Includes linear (y = mx + b), polynomial, and exponential functions.

Factoring

Process of breaking down expressions into simpler multiplicative forms (e.g., x² - 1 = (x - 1)(x + 1)).
Important for solving equations and simplifying mathematical expressions.

Systems of Equations

Involves multiple equations with several variables.
Solutions can be derived through:
- Substitution Method: Replace one variable with an expression from another equation.
- Elimination Method: Add or subtract equations to eliminate a variable.
- Graphical Method: Use graphical representations to find intersections that denote solutions.