Algebra Basics

Algebra

What is Algebra?

• A branch of mathematics that deals with the study of variables and their relationships
• Uses symbols, equations, and formulas to solve problems

Key Concepts:

Variables and Expressions

• Variables: letters or symbols that represent unknown values
• Expressions: combinations of variables, constants, and mathematical operations
• Example: 2x + 5 is an expression, where x is a variable and 2 and 5 are constants

Equations and Inequalities

• Equations: statements that two expressions are equal
• Inequalities: statements that one expression is greater than, less than, or equal to another
• Examples:
  • Equation: 2x + 3 = 5
  • Inequality: 2x - 4 > 3

Solving Equations and Inequalities

• Linear equations: equations in which the highest power of the variable is 1
• Quadratic equations: equations in which the highest power of the variable is 2
• Methods for solving equations:
  • Addition/Subtraction
  • Multiplication/Division
  • Factoring
  • Quadratic Formula

Graphing

• Graphing equations on a coordinate plane
• x-axis: horizontal axis
• y-axis: vertical axis
• Points on the graph represent solutions to the equation
• Example: graphing the equation y = 2x - 3

Functions

• Relations between a set of inputs (domain) and a set of possible outputs (range)
• Notation: f(x) = output
• Example: f(x) = 2x + 1, where x is the input and 2x + 1 is the output

Systems of Equations

• Sets of two or more equations that must be true at the same time
• Methods for solving systems:
  • Substitution
  • Elimination
  • Graphing

Polynomials

• Expressions consisting of variables and coefficients combined using only addition, subtraction, and multiplication
• Examples: x^2 + 3x - 4, 2x^3 - 5x^2 + x - 1

Algebra

Definition and Scope

• Algebra is a branch of mathematics that studies variables and their relationships
• It uses symbols, equations, and formulas to solve problems

Variables and Expressions

• Variables represent unknown values and are denoted by letters or symbols
• Expressions are combinations of variables, constants, and mathematical operations
• Example: 2x + 5 is an expression, where x is a variable and 2 and 5 are constants

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
• Examples:
  • Equation: 2x + 3 = 5
  • Inequality: 2x - 4 > 3

Solving Equations and Inequalities

• Linear equations have the highest power of the variable as 1
• Quadratic equations have the highest power of the variable as 2
• Methods for solving equations include:
  • Addition/Subtraction
  • Multiplication/Division
  • Factoring
  • Quadratic Formula

Graphing

• Graphing equations on a coordinate plane with x-axis (horizontal) and y-axis (vertical)
• Points on the graph represent solutions to the equation
• Example: graphing the equation y = 2x - 3

Functions

• Relations between a set of inputs (domain) and a set of possible outputs (range)
• Notation: f(x) = output
• Example: f(x) = 2x + 1, where x is the input and 2x + 1 is the output

Systems of Equations

• Sets of two or more equations that must be true at the same time
• Methods for solving systems include:
  • Substitution
  • Elimination
  • Graphing

Polynomials

• Expressions consisting of variables and coefficients combined using addition, subtraction, and multiplication
• Examples: x^2 + 3x - 4, 2x^3 - 5x^2 + x - 1