Equivalent expressions have the same value for every allowed input. Simplifying makes structure easier to read by distributing, combining like terms, and reducing common factors where the domain permits it.

A reliable method

  1. Distribute through each group.
  2. Keep the sign on every term.
  3. Combine only identical variable parts.
  4. Check equivalence by expansion; for fractions preserve original denominator restrictions.

Example 1

Simplify \(3 x + 5 x - 7\).

  1. Apply exponent rules or distribute through parentheses without dropping any signs.
  2. An expanded form is \(8 x - 7\).
  3. Combine only like terms and reduce common multiplicative factors: \(8 x - 7\).

Answer: \(8 x - 7\)

Example 2

Expand and collect like terms: \(2 \left(x + 4\right) - 3 \left(x - 1\right)\).

  1. Apply exponent rules or distribute through parentheses without dropping any signs.
  2. An expanded form is \(11 - x\).
  3. Combine only like terms and reduce common multiplicative factors: \(11 - x\).

Answer: \(11 - x\)

Common mistakes

  • Unlike terms such as x and x squared cannot be combined.
  • Cancellation applies to factors, not to terms joined by addition.

Practice without the answer first

  1. Simplify \(- 3 x^{2} + 4 x^{2} + 2 x + 6 x\).
  2. Expand and collect like terms: \(4 x - \left(2 x - 5\right)\).
  3. Simplify \(- 2 x + 6 x - 3 + 8\).
Answers and step-by-step explanations

Practice 1

  1. Apply exponent rules or distribute through parentheses without dropping any signs.
  2. An expanded form is \(x^{2} + 8 x\).
  3. Combine only like terms and reduce common multiplicative factors: \(x \left(x + 8\right)\).

Answer: \(x \left(x + 8\right)\)

Practice 2

  1. Apply exponent rules or distribute through parentheses without dropping any signs.
  2. An expanded form is \(4 x - \left(2 x - 5\right)\).
  3. Combine only like terms and reduce common multiplicative factors: \(4 x - \left(2 x - 5\right)\).

Answer: \(4 x - \left(2 x - 5\right)\)

Practice 3

  1. Apply exponent rules or distribute through parentheses without dropping any signs.
  2. An expanded form is \(4 x + 5\).
  3. Combine only like terms and reduce common multiplicative factors: \(4 x + 5\).

Answer: \(4 x + 5\)

Study context: This shared lesson supports math study across exams. Use it to repair a prerequisite or practice the matching skill; it is not an official SAT question set or a score prediction.

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