Algebra Basics Overview

Algebra Study Notes

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

• Basic Concepts:

  • Variables: Symbols (usually letters) that represent unknown values (e.g., x, y).
  • Constants: Fixed values (e.g., 2, -5).
  • Expressions: Combinations of variables and constants using operations (e.g., 3x + 2).
  • Equations: Statements that two expressions are equal (e.g., 2x + 3 = 7).

• Operations:

  • Addition, Subtraction, Multiplication, Division
  • Order of Operations: PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction).

• Solving Equations:

  1. Isolate the variable (e.g., add/subtract terms).
  2. Use inverse operations to solve for the variable.
  3. Check the solution by substituting back into the original equation.

• Types of Equations:

  • Linear Equations: Form y = mx + b, where m is the slope and b is the y-intercept.
  • Quadratic Equations: Form ax² + bx + c = 0, can be solved using factoring, completing the square, or the quadratic formula x = (-b ± √(b²-4ac))/(2a).
  • Polynomial Equations: Involves terms with variables raised to whole number exponents.

• Functions:

  • A relation where each input has exactly one output.
  • Notation: f(x), g(x), etc.
  • Types: Linear, Quadratic, Exponential, etc.

• Graphing:

  • Plotting points on a coordinate plane.
  • Understanding slopes (rise/run) for linear functions.
  • Identifying intercepts (where the graph crosses axes).

• Systems of Equations:

  • Set of two or more equations with the same variables.
  • Solutions can be found using:
    • Graphing
    • Substitution
    • Elimination

• Inequalities:

  • Similar to equations but use >, <, ≥, or ≤.
  • Solutions may include a range of values.

• Factoring:

  • Rewriting expressions as products of factors.
  • Common methods: factoring out the greatest common factor, using the difference of squares, or applying the quadratic formula.

• Applications:

  • Solving real-world problems involving relationships, rates, and quantities.
  • Used in fields such as engineering, economics, and natural sciences.

Algebra Overview

• Algebra involves symbols and rules for manipulating them to solve equations and express relationships.
• It serves as a foundational mathematical discipline applied in various fields.

Basic Concepts

• Variables: Symbols like x and y denote unknown values.
• Constants: Fixed numerical values, such as 2 or -5.
• Expressions: Combinations formed by variables and constants through operations (e.g., 3x + 2).
• Equations: Show that two expressions are equal (e.g., 2x + 3 = 7).

Operations

• Core operations include addition, subtraction, multiplication, and division.
• Order of Operations: Follow PEMDAS: Parentheses, Exponents, Multiplication and Division, Addition and Subtraction.

Solving Equations

• Steps to solve an equation include isolating the variable, using inverse operations, and checking the solution by substitution.

Types of Equations

• Linear Equations: Structured as y = mx + b, where m indicates slope and b signifies the y-intercept.
• Quadratic Equations: Formed as ax² + bx + c = 0, solvable through factoring, completing the square, or the quadratic formula x = (-b ± √(b²-4ac))/(2a).
• Polynomial Equations: Include terms with variable exponents that are whole numbers.

Functions

• Defined as relations where each input corresponds to one output.
• Notation typically includes f(x), g(x), etc., with various types like Linear, Quadratic, and Exponential.

Graphing

• Involves plotting points on a coordinate plane to visualize equations.
• Critical concepts include calculating slopes (rise/run) for linear functions and identifying intercepts (points where graphs cross axes).

Systems of Equations

• Comprise multiple equations sharing common variables.
• Solutions can be determined through graphing, substitution, or elimination methods.

Inequalities

• Similar to equations but express relationships using symbols like >, <, ≥, and ≤.