Equivalent Expressions: Factoring, Distributing, and Simplifying

🔑 Key Concepts

  • Equivalent expressions have the same value even when they are written differently.
  • The distributive property can be used to expand or factor an expression.
  • Factoring out the greatest common factor rewrites an expression as a product.
  • Like terms can be combined only when they have identical variable parts.
  • Real-world situations may have several equivalent algebraic representations.
  • Negative signs must be distributed to every term inside parentheses.
  • Substitution can be used to test whether two expressions are equivalent.

✏️ Worked Examples — Equivalent Expressions Review

🧠 Math Vocabulary

  • Equivalent expressions: Expressions that have the same value for every possible value of their variables.
  • Distributive property: Multiplying a factor by every term inside parentheses.
  • Factor: A number or expression multiplied by another number or expression.
  • Greatest common factor: The largest factor shared by every term in an expression.
  • Like terms: Terms that have identical variable parts.
  • Coefficient: The numerical factor multiplying a variable.
  • Constant: A term without a variable.
  • Substitution: Replacing a variable with a value to evaluate or compare expressions.

💡 Main Idea

Equivalent expressions can be created by distributing, combining like terms, factoring, or rewriting a real-world relationship. Expressions do not need to look identical to represent the same quantity. For example, \(5(n-3)\), \(5n-15\), and \(4(n-3)+(n-3)\) are equivalent because each simplifies to the same expression.

📚 What You Should Already Know

Before this lesson, students should understand integer and fraction operations, coefficients, terms, factors, and like terms. Students should also know how to apply the distributive property and translate basic verbal or geometric relationships into algebraic expressions.

🚀 What Comes Next

Next, students can use equivalent expressions when solving equations, working with inequalities, evaluating formulas, and analyzing linear relationships. These skills also support algebraic modeling in geometry, percent situations, functions, and real-world applications.

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