Algebra Basics

Algebra

Definition

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

Types of Algebra

• Elementary Algebra: deals with the basic concepts of algebra, including linear equations, quadratic equations, and functions.
• Intermediate Algebra: builds upon elementary algebra, covering topics such as systems of equations, inequalities, and functions.
• College Algebra: covers advanced topics, including polynomial and rational functions, exponential and logarithmic functions, and series and sequences.

Key Concepts

• Variables and Expressions:
  • Variables: letters or symbols that represent unknown values or quantities.
  • Expressions: combinations of variables, constants, and mathematical operations.
• Equations and Inequalities:
  • Equations: statements that two expressions are equal.
  • Inequalities: statements that one expression is greater than, less than, or equal to another.
• Functions:
  • Domain: the set of input values for a function.
  • Range: the set of output values for a function.
  • Composition of functions: combining two or more functions to create a new function.

Algebraic Operations

• Addition and Subtraction:
  • Combining like terms: combining terms with the same variable and coefficient.
  • Combining unlike terms: combining terms with different variables or coefficients.
• Multiplication and Division:
  • Distributive property: multiplying a single value to multiple terms.
  • Factoring: expressing an expression as a product of simpler expressions.

Graphing and Solving

• Graphing:
  • Cartesian coordinate system: a system of axes used to graph equations.
  • Graphing linear equations: using the slope-intercept form (y = mx + b) to graph lines.
• Solving Equations and Inequalities:
  • Linear equations: solving for x in equations of the form ax + by = c.
  • Quadratic equations: solving for x in equations of the form ax^2 + bx + c = 0.

Algebra

Definition

• Algebra is a branch of mathematics dealing with variables and their relationships, expressed through symbols, equations, and functions.

Types of Algebra

• Elementary Algebra covers linear equations, quadratic equations, and functions.
• Intermediate Algebra builds upon elementary algebra, covering systems of equations, inequalities, and functions.
• College Algebra covers advanced topics like polynomial and rational functions, exponential and logarithmic functions, and series and sequences.

Key Concepts

Variables and Expressions

• Variables are letters or symbols representing unknown values or quantities.
• Expressions are combinations of variables, constants, and mathematical operations.

Equations and Inequalities

• Equations are statements that two expressions are equal.
• Inequalities are statements that one expression is greater than, less than, or equal to another.

Functions

• Domain is the set of input values for a function.
• Range is the set of output values for a function.
• Composition of functions combines two or more functions to create a new function.

Algebraic Operations

Addition and Subtraction

• Combining like terms combines terms with the same variable and coefficient.
• Combining unlike terms combines terms with different variables or coefficients.

Multiplication and Division

• Distributive property multiplies a single value to multiple terms.
• Factoring expresses an expression as a product of simpler expressions.

Graphing and Solving

Graphing

• Cartesian coordinate system is a system of axes used to graph equations.
• Graphing linear equations uses the slope-intercept form (y = mx + b) to graph lines.

Solving Equations and Inequalities

• Linear equations solve for x in equations of the form ax + by = c.
• Quadratic equations solve for x in equations of the form ax^2 + bx + c = 0.