Introduction to Variables and Expressions

Mar 20, 2025

Algebra 1: Lesson 1 - Introduction to Variables and Expressions

Key Concepts

  • Variables: A variable is a symbol, usually a lowercase letter, used to represent an unknown quantity. It acts as a placeholder. Common variables in algebra include x, y, and z, but any letter can be used.

  • Expressions: These are combinations of numbers, variables, and operators (such as +, -, *, /) that represent a mathematical relationship.

Example Problems

  1. Daily Earnings Example:

    • Mark earns $63 per day plus tips.
    • Expression: $63 + x$, where x is the tips.
    • If tips are $7, then total earnings = $63 + $7 = $70.

  2. Cost of Milk Example:

    • Cost for 2 gallons of milk is represented as $2y$, where y is the cost per gallon.
    • If 1 gallon costs $1, then cost for 2 gallons = $2 * $1 = $2.

Terms and Definitions

  • Term: A single number, a single variable, or a number multiplied by one or more variables.

    • Examples: $4x$, $9y$, $24xyz$, $-3a$.

  • Coefficient: The number multiplying the variable in a term.

    • Example: In $4x^2z$, 4 is the coefficient.

  • Constant: A number on its own without any variables.

    • Example: In $5x + 7$, 7 is the constant.

  • Algebraic Expression: One or more terms separated by + or - signs.

    • Example: $5x + 2y - 4z$.

Simplifying Algebraic Expressions

  • Use the distributive property to simplify expressions:

    • Example: $3(x - 4) = 3x - 12$.

  • Combine like terms:

    • Example: $2x + 3x = 5x$.

Practice Problems

  1. Simplify $3(x - 4)$

    • Solution: $3x - 12$

  2. Combine like terms: $2x + 3x$

    • Solution: $5x$

Visualizing Equations

  • Algebraic Expression vs. Equation:
    • Expression: Mathematical phrase without an equal sign.
    • Equation: Mathematical statement showing equality, includes an equal sign (e.g., $2x + 3 = 11$).

Solving Equations

  • Basic concept: To find the value of the variable that makes the equation true.
  • Example: $2x + 3 = 11$
    • Solution: Subtract 3 from both sides, $2x = 8$, then divide by 2, $x = 4$.

Conclusion

  • Understanding variables and expressions is fundamental in algebra.
  • Practice simplifying expressions and solving simple equations to build a strong foundation for more complex algebraic concepts.