Solving Multi-Step Equations with Variables on Both Sides

🔑 Key Concepts

  • Use the distributive property to remove parentheses before solving.
  • Distribute negative factors carefully to every term inside the parentheses.
  • Combine like terms on each side of the equation.
  • Move variable terms to one side and constants to the other.
  • Perform the same operation on both sides to keep the equation balanced.
  • Check the solution by substituting it into the original equation.

✏️ Worked Examples — Equations with Distribution and Variables on Both Sides

🧠 Math Vocabulary

  • Distributive property: Multiplying a factor by every term inside parentheses.
  • Like terms: Terms containing the same variable raised to the same power.
  • Variable term: A term containing a variable, such as \(8x\) or \(-12x\).
  • Constant: A number without a variable, such as \(-14\) or \(36\).
  • Inverse operations: Operations that undo one another and help isolate a variable.
  • Equivalent equations: Equations that have the same solution.
  • Isolate: To get the variable alone on one side of an equation.

💡 Main Idea

This lesson focuses on solving multi-step equations that require distribution, combining like terms, and moving variable terms between both sides of the equation. Special attention is needed when distributing negative factors because every term inside the parentheses changes according to multiplication rules. After both sides are simplified, inverse operations are used to isolate the variable.

📚 What You Should Already Know

Before this lesson, students should understand integer multiplication, the distributive property, combining like terms, and solving one-step and two-step equations. Students should also know how performing the same operation on both sides preserves equality.

🚀 What Comes Next

Next, students can analyze equations that result in one solution, no solution, or infinitely many solutions. These equation-solving strategies also prepare students for inequalities, systems of equations, and algebraic modeling.

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