Solving Equations With Variables on Both Sides

🔑 Key Concepts

  • When variables appear on both sides, move the variable terms to one side and constants to the other.
  • Use the distributive property before combining like terms.
  • Equations containing fractions can be simplified by multiplying both sides by the least common denominator.
  • Equations with one fraction on each side can also be rewritten using cross multiplication.
  • 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 Variables, Distribution, and Fractions

🧠 Math Vocabulary

  • Variable term: A term that contains a variable, such as \(3a\) or \(-2a\).
  • Constant: A number without a variable, such as \(-22\) or \(-7\).
  • Distributive property: Multiplying a factor by every term inside parentheses.
  • Least common denominator: The smallest common multiple of all denominators in an equation.
  • Cross multiplication: Multiplying each numerator by the denominator on the opposite side of a proportion.
  • Inverse operations: Operations that undo one another and help isolate a variable.

💡 Main Idea

This lesson focuses on solving linear equations involving variables on both sides, distribution, proportions, and multiple fractions. The most efficient first step depends on the structure of the equation. Students may distribute, combine like terms, cross multiply, or multiply by the least common denominator before using inverse operations to isolate the variable.

📚 What You Should Already Know

Before this lesson, students should know how to combine like terms, use the distributive property, solve one-step and two-step equations, and perform operations with fractions. Students should also understand equivalent equations and how performing the same operation on both sides preserves equality.

🚀 What Comes Next

Next, students can apply these strategies to equations with more complicated rational expressions, equations with special solution cases, inequalities, formulas, and algebraic word problems. These skills also prepare students for systems of equations and more advanced algebra.

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