Equations and Variables in Algebra

Equations

• A statement that says two mathematical expressions are equal
• Typically written with an equal sign (=) separating the two expressions
• Examples:
  • 2x + 3 = 5
  • x^2 - 4 = 0
• Can be:
  • Simple equations: only one variable and no exponents (e.g., 2x = 4)
  • Linear equations: degree of the variable is 1 (e.g., 2x + 3 = 5)
  • Quadratic equations: degree of the variable is 2 (e.g., x^2 - 4 = 0)

Variables

• A symbol that represents a value that can change
• Often represented by letters (e.g., x, y, z)
• Can be:
  • Independent variable: changes freely (e.g., x in the equation y = 2x)
  • Dependent variable: changes in response to the independent variable (e.g., y in the equation y = 2x)
• Variables can be:
  • Discrete: takes on specific, distinct values (e.g., whole numbers)
  • Continuous: takes on any value within a certain range or interval (e.g., real numbers)
• Importance of variables:
  • Allow us to generalize and model real-world situations
  • Enable us to solve problems and make predictions

Equations

• Defined as a statement that equates two mathematical expressions
• Typically denoted by an equal sign (=) separating the two expressions
• Examples of equations include:
  • Simple equations with one variable and no exponents (e.g., 2x = 4)
  • Linear equations with a degree of 1 (e.g., 2x + 3 = 5)
  • Quadratic equations with a degree of 2 (e.g., x^2 - 4 = 0)

Variables

• Represented by symbols, often letters (e.g., x, y, z)
• Can be independent or dependent
  • Independent variables change freely (e.g., x in y = 2x)
  • Dependent variables change in response to the independent variable (e.g., y in y = 2x)
• Variables can be:
  • Discrete, taking on specific, distinct values (e.g., whole numbers)
  • Continuous, taking on any value within a certain range or interval (e.g., real numbers)
• Importance of variables:
  • Allow generalization and modeling of real-world situations
  • Enable problem-solving and prediction