Algebra Concepts and Operations

Algebra

Definition: A branch of mathematics dealing with symbols and the rules for manipulating those symbols to solve equations.
Key Concepts:
- Variables: Symbols (often letters) used to represent unknown values.
- Constants: Fixed values that do not change.
- Expressions: Combinations of variables, constants, and operators (e.g., 3x + 5).
- Equations: Mathematical statements that assert the equality of two expressions (e.g., 2x + 3 = 7).
Operations:
- Addition, subtraction, multiplication, and division of algebraic expressions.
- Factoring: Breaking down an expression into simpler components (e.g., x² - 9 = (x - 3)(x + 3)).
- Distributive Property: a(b + c) = ab + ac.
Types of Equations:
- Linear Equations: Represented as y = mx + b (where m is the slope and b is the y-intercept).
- Quadratic Equations: Standard form ax² + bx + c = 0, can be solved using factoring, completing the square, or the quadratic formula.
- Polynomial Equations: Involves terms with variables raised to whole number powers.
Functions:
- Definition: A relation that assigns exactly one output for each input.
- Types of functions: Linear, quadratic, exponential, and logarithmic.
- Function notation: f(x) represents the function f evaluated at x.
Systems of Equations:
- Definition: A set of two or more equations with the same variables.
- Methods to solve:
  - Graphical method: Plotting equations on a graph.
  - Substitution: Solving one equation for a variable and substituting it into the other.
  - Elimination: Adding or subtracting equations to eliminate a variable.
Inequalities:
- Similar to equations but use inequality symbols (>, <, ≥, ≤).
- Solutions can be represented on a number line.
- System of inequalities: Multiple inequalities that must hold true simultaneously.
Applications:
- Used in diverse fields such as engineering, physics, finance, and statistics.
- Problem-solving and modeling real-world scenarios.
Graphing:
- Coordinate plane: Consists of x-axis and y-axis.
- Plotting points: (x, y) coordinates to represent solutions.
- Graphing linear equations involves finding intercepts and drawing a straight line through them.
Key Formulas:
- Slope formula: m = (y2 - y1) / (x2 - x1).
- Quadratic formula: x = (-b ± √(b² - 4ac)) / (2a).
- Distance formula: d = √((x2 - x1)² + (y2 - y1)²).

Algebra Overview

A branch of mathematics focused on symbols and rules for solving equations.

Key Concepts

Variables: Represent unknown values, commonly denoted by letters.
Constants: Fixed values that remain unchanged in expressions and equations.
Expressions: Formed by combining variables, constants, and operators (example: 3x + 5).
Equations: Statements asserting equality between two expressions (example: 2x + 3 = 7).

Operations

Involves addition, subtraction, multiplication, and division of algebraic expressions.
Factoring: Simplifying expressions into products of simpler terms (example: x² - 9 = (x - 3)(x + 3)).
Distributive Property: Describes how to multiply a single term by a sum (expressed as a(b + c) = ab + ac).

Types of Equations

Linear Equations: Expressed in the form y = mx + b, where m is the slope and b is the y-intercept.
Quadratic Equations: Formulated as ax² + bx + c = 0 and solved through methods like factoring, completing the square, or using the quadratic formula.
Polynomial Equations: Involves terms with variables raised to whole number exponents.

Functions

Defined as relations that associate one specific output with each input.
Categorized into types such as linear, quadratic, exponential, and logarithmic.
Function Notation: Expressed as f(x), which indicates the function evaluated at x.

Systems of Equations

Comprise two or more equations sharing the same variables.
Solving Methods:
- Graphical Method: Involves plotting the equations on a coordinate graph.
- Substitution: Involves isolating a variable in one equation and substituting back into the other.
- Elimination: Achieves variable elimination by adding or subtracting equations from one another.