Algebra Concepts and Operations

Algebra

Definition: A branch of mathematics dealing with symbols and the rules for manipulating those symbols; represents numbers and quantities in formulas and equations.
Key Concepts:
- Variables: Symbols (often letters) used to represent unknown values (e.g., x, y).
- Constants: Fixed values that do not change (e.g., 5, -3).
- Expressions: Combinations of variables and constants using mathematical operations (e.g., 2x + 3).
- Equations: A statement that two expressions are equal (e.g., 2x + 3 = 7).
Operations:
- Addition, subtraction, multiplication, division.
- Order of operations (PEMDAS/BODMAS): Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right).
Solving Equations:
- Isolate the variable: Use inverse operations to solve for the unknown (e.g., if 2x + 3 = 7, subtract 3 and then divide by 2).
- Linear equations: Equations of the first degree, such as y = mx + b, where m is the slope and b is the y-intercept.
- Quadratic equations: Equations of the second degree in the form ax^2 + bx + c = 0; solutions can be found using factoring, completing the square, or the quadratic formula.
Functions:
- Definition: A relation between a set of inputs and a set of permissible outputs, typically expressed as f(x).
- Types of functions:
  - Linear: Graphs as straight lines; represented in the form f(x) = mx + b.
  - Quadratic: Graphs as parabolas; represented in the form f(x) = ax^2 + bx + c.
  - Polynomial: Expressions that involve variables raised to whole number powers.
  - Exponential: Functions of the form f(x) = ab^x, where b is a positive constant.
Inequalities:
- Statements about the relative size or order of two values; expressed using symbols like <, >, ≤, ≥.
- Solutions can be represented on a number line or in interval notation.
Graphing:
- Coordinate System: A plane defined by x (horizontal) and y (vertical) axes.
- Plotting Points: Represent coordinates (x, y) on the graph.
- Slope: Measure of steepness of the line; calculated as rise over run (change in y over change in x).
Factoring:
- Breaking down expressions into products of simpler expressions (e.g., factoring x^2 - 9 into (x + 3)(x - 3)).
- Common methods include grouping, using the distributive property, and applying special products (e.g., difference of squares).
Applications:
- Used in problem-solving across disciplines, including science, economics, engineering, and statistics.
- Fundamental for understanding higher-level mathematics, such as calculus and linear algebra.

Algebra

Definition: Algebra is a branch of mathematics that deals with symbols and the rules for manipulating those symbols. It uses symbols called variables to represent unknown values and constants to represent fixed values.
Key Concepts:
- Variables: Letters that represent unknown values.
- Constants: Numbers that have fixed values.
- Expressions: Combinations of variables and constants using math operations.
- Equations: Statements where two expressions are equal.
- Operations: Addition, subtraction, multiplication, and division.
- Order of Operations (PEMDAS/BODMAS): Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right).

Solving Equations

Isolate the Variable: Use inverse operations to get the variable by itself. For example, if 2x + 3 = 7, subtract 3 from both sides, then divide by 2 to find x.
Linear Equations: Equations in the form y = mx + b, where m is the slope and b is the y-intercept. Their graphs are straight lines.
Quadratic Equations: Equations in the form ax^2 + bx + c = 0. Solutions can be found by factoring, completing the square, or using the quadratic formula.

Functions

Definition: A relation between a set of inputs and their corresponding outputs.
Examples:
- Linear Functions: Graph as straight lines, represented by f(x) = mx + b.
- Quadratic Functions: Graph as parabolas, represented by f(x) = ax^2 + bx + c.
- Polynomial Functions: Expressions that involve variables raised to whole number powers.
- Exponential Functions: Functions of the form f(x) = ab^x, where b is a positive constant.

Inequalities

Definition: Statements about the relative size or order of two values.
Symbols: <, >, ≤, ≥.
Solutions: Can be represented on a number line or using interval notation.

Graphing

Coordinate System: Uses x (horizontal) and y (vertical) axes to represent a plane.
Plotting Points: (x, y) coordinates are used to plot points on the graph.
Slope: The steepness of a line, calculated by rise over run (change in y over change in x).

Factoring

Definition: Breaking down expressions into simpler expressions (products).
Methods:
- Grouping: Similar terms are grouped together to factor.
- Distributive Property: Used to expand expressions.
- Special Products: Formulas for differences of squares or sums of cubes.

Applications

Problem Solving: Used in various fields like science, economics, engineering, and statistics.
Foundation for Higher-Level Math: Essential for understanding calculus and linear algebra.