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Understanding Variables and Expressions in Algebra

Apr 25, 2025

Algebra One Lecture: Variables and Expressions

Introduction

  • Transition from pre-algebra to algebra.
  • Introduction to variables and expressions.

Variables

  • Definition: A variable is a symbol (generally a lowercase letter) that represents an unknown quantity.
  • Common examples: x, y, z.
  • Usage: Variables act as placeholders for various values.

Expressions

  • Definition: An expression is a mathematical phrase that can include numbers, variables, and operation symbols.
  • Example: Mark earns $63 per day plus tips, represented as $63 + x$, where $x$ is the amount in tips.
  • Calculating Example: If Mark received $7 in tips, the expression becomes $63 + 7 = $70.

Modeling Situations

  • Example: Jason buys two gallons of milk at an unknown price per gallon. Expression: $2y$, where $y$ is the price per gallon.

Terms

  • Definition: A term can be a single number, a variable, or a number times a variable.
  • Examples: 4x, 9y, 24xyz, -3a.
  • Notes: Variables in terms can be raised to powers.

Coefficients

  • Definition: A coefficient is a numerical factor multiplying a variable.
  • Example: In 4x, 4 is the coefficient.

Constants

  • Definition: A constant is a fixed value that does not change.
  • Example: In the expression 5x + 7, 7 is the constant.

Algebraic Expressions

  • Definition: One or more terms separated by plus or minus symbols.
  • Example: 5x + 2y - 4z includes terms 5x, 2y, and 4z.
  • Note: The value of an expression can change with different values for the variables.

Evaluation of Expressions

  • Plug values into expressions:
    • Example: $2x + 3$, where $x = 4$, results in $2(4) + 3 = 11$.

Distributive Property

  • Definition: Multiplying a sum by a number gives the same result as multiplying each addend individually and then adding the products.
  • Example: 3 times (x - 4) simplifies to 3x - 12.

Like Terms

  • Definition: Terms with the same variable raised to the same power.
  • Examples: 5x and 3x are like terms, but 5x and 3y are not.

Combining Like Terms

  • Add or subtract the coefficients of like terms while keeping the variable part the same.
  • Example: $2x + 3x = 5x$.

Simplification

  • Simplifying algebraic expressions using properties like distributive property and combining like terms.
  • Example: $-2(4x - 7) + 4(x - 3)$ simplifies as $-8x + 14 + 4x - 12 = -4x + 2$.

These notes cover the foundational concepts of algebra introduced in the lecture, including variables, expressions, terms, coefficients, constants, and basic operations with expressions. They provide a clear guide for understanding and working with algebraic expressions.

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