Algebra Basics

Algebra

Definition: A branch of mathematics dealing with symbols and the rules for manipulating those symbols to solve equations and represent relationships.
Key Concepts:
- Variables: Symbols (often letters) used to represent unknown values.
- Constants: Fixed values that do not change.
- Expressions: Combinations of variables and constants using operations (e.g., 3x + 2).
- Equations: Mathematical statements that show two expressions are equal (e.g., 2x + 3 = 7).
Operations:
- Addition, Subtraction, Multiplication, Division: Fundamental operations applied to variables and constants.
- Order of Operations: PEMDAS/BODMAS (Parentheses/Brackets, Exponents/Orders, Multiplication and Division, Addition and Subtraction).
Types of Algebra:
- Elementary Algebra: Basics of algebraic operations, solving simple equations.
- Abstract Algebra: Study of algebraic structures like groups, rings, and fields.
- Linear Algebra: Focus on vector spaces and linear mappings between them.
Solving Equations:
- Linear Equations: Equations of the form ax + b = c. Solution involves isolating the variable.
- Quadratic Equations: Equations of the form ax² + bx + c = 0. Solved using factoring, completing the square, or the quadratic formula.
Functions:
- Definition: A relation where each input (x) has a single output (y).
- Notation: f(x) represents a function of x.
- Types: Linear, quadratic, polynomial, exponential, logarithmic.
Graphing:
- Coordinate Plane: Grid defined by x-axis (horizontal) and y-axis (vertical).
- Graph of Linear Functions: Straight line, described by slope (m) and y-intercept (b) in the form y = mx + b.
Factoring:
- Definition: Expressing an expression as a product of its factors.
- Common Methods: Factoring out the greatest common factor, using special products (difference of squares, perfect square trinomials).
Inequalities:
- Definition: Mathematical statements indicating that one expression is greater or less than another.
- Notation: Symbols include >, <, ≥, ≤.
- Graphing: Solutions represented on a number line or coordinate plane, often using shaded regions.
Exponents and Radicals:
- Exponents: Indicate repeated multiplication (e.g., x² = x * x).
- Laws of Exponents: Include product of powers, power of a power, quotient of powers.
- Radicals: Expressions involving roots (e.g., √x), where x is a non-negative number.
Applications:
- Modeling Real-World Problems: Using algebra to formulate and solve problems in fields such as physics, engineering, economics.
- Systems of Equations: Solving for multiple variables simultaneously, using methods like substitution or elimination.

Algebra

Definition: Algebra is a branch of mathematics dealing with symbols and rules for manipulating them to solve equations and represent relationships.

Key Concepts

Variables: Symbols (often letters) used to represent unknown values.
Constants: Fixed values that do not change.
Expressions: Combinations of variables and constants using operations (e.g., 3x + 2).
Equations: Mathematical statements that show two expressions are equal (e.g., 2x + 3 = 7).

Operations

Addition, Subtraction, Multiplication, Division: Fundamental operations applied to variables and constants.
Order of Operations: Follow PEMDAS/BODMAS (Parentheses/Brackets, Exponents/Orders, Multiplication and Division, Addition and Subtraction).

Types of Algebra

Elementary Algebra: Studies basics of algebraic operations, solving simple equations.
Abstract Algebra: Explores algebraic structures like groups, rings, and fields.
Linear Algebra: Focuses on vector spaces and linear mappings between them.

Solving Equations

Linear Equations: Equations of the form ax + b = c, solved by isolating the variable.
Quadratic Equations: Equations of the form ax² + bx + c = 0, solved using factoring, completing the square, or the quadratic formula.

Functions

Definition: A relation where each input (x) has a single output (y).
Notation: f(x) represents a function of x.
Types: Linear, quadratic, polynomial, exponential, logarithmic functions.

Graphing

Coordinate Plane: Grid defined by x-axis (horizontal) and y-axis (vertical).
Graph of Linear Functions: Straight line, described by slope (m) and y-intercept (b) in the form y = mx + b.

Factoring

Definition: Expressing an expression as a product of its factors.
Common Methods: Factoring out the greatest common factor, using special products (difference of squares, perfect square trinomials).