Algebra Basics

Algebra

Definition

• Algebra is a branch of mathematics that deals with variables and their relationships, often expressed through the use of symbols, equations, and functions.

Key Concepts

• Variables: Letters or symbols that represent unknown values or quantities.
• Constants: Numbers that do not change value.
• Algebraic Expressions: Combinations of variables, constants, and mathematical operations.
• Equations: Statements that express the equality of two algebraic expressions.

Types of Equations

• Linear Equations: Equations in which the highest power of the variable(s) is 1.
• Quadratic Equations: Equations in which the highest power of the variable(s) is 2.
• Polynomial Equations: Equations involving variables and coefficients of whole-number exponents.

Solving Equations

• Addition and Subtraction Properties: Adding or subtracting the same value to both sides of an equation does not change its solution.
• Multiplication and Division Properties: Multiplying or dividing both sides of an equation by a non-zero value does not change its solution.
• Inverse Operations: Reversing the operation to isolate the variable (e.g., adding the opposite to both sides to eliminate a term).

Functions

• Domain: The set of input values for which a function is defined.
• Range: The set of output values of a function.
• Composition of Functions: Combining two or more functions to create a new function.

Graphing

• Coordinate Plane: A two-dimensional plane with x and y axes used to graph algebraic equations.
• X-Intercept: The point at which a graph crosses the x-axis.
• Y-Intercept: The point at which a graph crosses the y-axis.
• Slope-Intercept Form: The equation of a line in the form y = mx + b, where m is the slope and b is the y-intercept.

Algebra

Definition

• Deals with variables and their relationships using symbols, equations, and functions.

Key Concepts

Variables and Constants

• Variables: represent unknown values or quantities, denoted by letters or symbols.
• Constants: numbers with fixed values, do not change.

Algebraic Expressions and Equations

• Algebraic Expressions: combinations of variables, constants, and mathematical operations.
• Equations: statements expressing equality of two algebraic expressions.

Types of Equations

Linear, Quadratic, and Polynomial Equations

• Linear Equations: highest power of variables is 1, e.g., 2x + 3 = 5.
• Quadratic Equations: highest power of variables is 2, e.g., x^2 + 4x + 4 = 0.
• Polynomial Equations: involve variables and coefficients of whole-number exponents.

Solving Equations

Properties and Inverse Operations

• Addition and Subtraction Properties: adding or subtracting same value to both sides doesn't change solution.
• Multiplication and Division Properties: multiplying or dividing both sides by non-zero value doesn't change solution.
• Inverse Operations: reversing operations to isolate variables, e.g., adding opposites to eliminate terms.

Functions

Domain, Range, and Composition

• Domain: set of input values for which a function is defined.
• Range: set of output values of a function.
• Composition of Functions: combining two or more functions to create a new function.

Graphing

Coordinate Plane and Intercepts

• Coordinate Plane: two-dimensional plane with x and y axes for graphing algebraic equations.
• X-Intercept: point where graph crosses x-axis.
• Y-Intercept: point where graph crosses y-axis.
• Slope-Intercept Form: equation of a line in form y = mx + b, where m is slope and b is y-intercept.