Class Note: Variables, Expressions, and Equations

What is a Variable?

• A variable is a letter (often x) that represents a number with a value.

Expression vs. Equation

Algebraic Expression

• Does not have an equal sign (=).
• Contains variables (e.g., x).
• The variable in an expression can have multiple values.
• Example: If the expression is x + 5:
  • If x = 2, then value = 2 + 5 = 7
  • If x = 7, then value = 7 + 5 = 12
  • If x = -4, then value = -4 + 5 = 1
• The value of the expression changes as the variable changes.

Equation

• Has an equal sign (=).
• Contains variables, but the variable(s) must satisfy the equation.
• The variable in an equation has specific value(s) that make the equation true.
  • Example: x + 5 = 8
    • Only x = 3 will satisfy the equation since 3 + 5 = 8.
  • For linear equations (like the one above), there is one specific value.
  • For quadratic equations (e.g., x² + bx + c = 0), there can be two specific values.
• Equations are used to find which number makes the equation true.

Key Differences

| | Expressions | Equations | |------------------------|------------------------------|------------------------------------| | Equal Sign Present?| No | Yes | | Variable Values | Multiple possible values | Specific value(s) | | Purpose | Shows relationship/calculation| Solves for unknown(s) |

Summary

• Variables represent numbers and can take different values in expressions.
• Equations use variables, but only certain values solve the equation.
• Knowing the difference helps in solving math problems and understanding algebra.