Algebra Basics

Algebra

What is Algebra?

• 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 and Expressions

• A variable is a symbol that represents a value that can change
• An algebraic expression is a combination of variables, constants, and mathematical operations
• Examples: 2x, 3x^2, x + 5

Equations and Inequalities

• An equation is a statement that says two expressions are equal
• A solution to an equation is a value of the variable that makes the equation true
• Inequalities are statements that compare two expressions using greater than, less than, or equal to

Linear Equations

• A linear equation is an equation in which the highest power of the variable(s) is 1
• Linear equations can be written in the form: ax + by = c, where a, b, and c are constants
• Linear equations can be solved using addition, subtraction, multiplication, and division

Quadratic Equations

• A quadratic equation is an equation in which the highest power of the variable(s) is 2
• Quadratic equations can be written in the form: ax^2 + bx + c = 0, where a, b, and c are constants
• Quadratic equations can be solved using factoring, the quadratic formula, or graphing

Functions

• A function is a relation between a set of inputs (called the domain) and a set of possible outputs (called the range)
• Functions can be represented algebraically, graphically, or numerically
• Functions can be composed, added, and multiplied

Systems of Equations

• A system of equations is a set of two or more equations that must be true at the same time
• Systems of equations can be solved using substitution, elimination, or graphing
• Systems of equations can have no solution, one solution, or infinitely many solutions

Overview of Algebra

• A branch of mathematics focused on variables and their relationships.
• Involves the study of symbols and rules for manipulating these symbols to solve equations.

Key Concepts

Variables and Expressions

• A variable represents a value that can change, typically shown as letters (e.g., x, y).
• An algebraic expression includes variables, constants, and operations (e.g., addition, subtraction).
• Examples of expressions:
  • Single variable: 2x
  • Polynomial: 3x^2
  • Combined: x + 5

Equations and Inequalities

• An equation equates two expressions, indicated by an equal sign.
• The solution to an equation identifies the variable value that satisfies the equation.
• Inequalities express a relationship using symbols like > (greater than), < (less than), ≥ (greater than or equal to).

Linear Equations

• A linear equation has a maximum variable power of 1.
• Common form: ax + by = c, where a, b, and c are constants.
• Can be solved through basic arithmetic operations: addition, subtraction, multiplication, division.

Quadratic Equations

• A quadratic equation features a maximum variable power of 2.
• Standard form: ax^2 + bx + c = 0, with a, b, and c as constants.
• Solutions can be found through factoring, the quadratic formula, or graphing methods.

Functions

• A function links a set of inputs (domain) with outputs (range).
• Representation methods include algebraic expressions, graphical plots, and numerical tables.
• Functions can undergo operations such as composition, addition, and multiplication.

Systems of Equations

• A system consists of two or more equations that must hold true simultaneously.
• Methods for solving include substitution, elimination, and graphical representation.
• Solutions vary: there may be no solutions, one solution, or infinitely many solutions.