Introduction to Algebra

Algebra

Definition

• Algebra is a branch of mathematics that deals with variables and their relationships.
• It involves the study of mathematical symbols and the rules for manipulating these symbols to solve equations and study structures.

Key Concepts

• Variables: Symbols that represent unknown values or quantities.
• Constants: Symbols that represent known values or quantities.
• Algebraic expressions: Combinations of variables, constants, and mathematical operations.
• Equations: Statements that two algebraic expressions are equal.
• Inequalities: Statements that one algebraic expression is greater than, less than, or equal to another.

Types of Equations

• Linear equations: Equations in which the highest power of the variable(s) is 1.
  • Example: 2x + 3 = 5
• Quadratic equations: Equations in which the highest power of the variable(s) is 2.
  • Example: x^2 + 4x + 4 = 0
• Polynomial equations: Equations in which the variables are raised to non-negative integer powers.
  • Example: x^3 - 2x^2 - 5x + 1 = 0

Solving Equations

• Addition/Subtraction method: Add or subtract the same value to both sides of the equation to isolate the variable.
• Multiplication/Division method: Multiply or divide both sides of the equation by the same non-zero value to isolate the variable.
• Graphical method: Use graphs to find the intersection point(s) of the equation, which represents the solution.

Graphing

• Coordinate plane: A two-dimensional plane with a horizontal x-axis and a vertical y-axis.
• Graph of an equation: The set of all points that satisfy the equation.
• X-intercept: The point where the graph intersects the x-axis.
• Y-intercept: The point where the graph intersects the y-axis.

Functions

• Domain: The set of all input values of a function.
• Range: The set of all output values of a function.
• Function notation: f(x) is used to denote a function of x.
• Composition of functions: The result of applying one function to the output of another function.

Algebra

Definition of Algebra

• Algebra is a branch of mathematics dealing with variables and their relationships, involving study of mathematical symbols and rules for manipulating these symbols to solve equations and study structures.

Key Concepts in Algebra

Variables and Constants

• Variables: Symbols representing unknown values or quantities.
• Constants: Symbols representing known values or quantities.

Algebraic Expressions, Equations, and Inequalities

• Algebraic expressions: Combinations of variables, constants, and mathematical operations.
• Equations: Statements that two algebraic expressions are equal.
• Inequalities: Statements that one algebraic expression is greater than, less than, or equal to another.

Types of Equations in Algebra

Linear Equations

• Linear equations: Equations in which the highest power of the variable(s) is 1, e.g., 2x + 3 = 5.

Quadratic Equations

• Quadratic equations: Equations in which the highest power of the variable(s) is 2, e.g., x^2 + 4x + 4 = 0.

Polynomial Equations

• Polynomial equations: Equations in which the variables are raised to non-negative integer powers, e.g., x^3 - 2x^2 - 5x + 1 = 0.

Solving Equations in Algebra

Methods for Solving Equations

• Addition/Subtraction method: Add or subtract the same value to both sides of the equation to isolate the variable.
• Multiplication/Division method: Multiply or divide both sides of the equation by the same non-zero value to isolate the variable.
• Graphical method: Use graphs to find the intersection point(s) of the equation, which represents the solution.

Graphing in Algebra

Coordinate Plane

• Coordinate plane: A two-dimensional plane with a horizontal x-axis and a vertical y-axis.

Graph of an Equation

• Graph of an equation: The set of all points that satisfy the equation.

Intercepts

• X-intercept: The point where the graph intersects the x-axis.
• Y-intercept: The point where the graph intersects the y-axis.

Functions in Algebra

Domain and Range

• Domain: The set of all input values of a function.
• Range: The set of all output values of a function.

Function Notation and Composition

• Function notation: f(x) is used to denote a function of x.
• Composition of functions: The result of applying one function to the output of another function.