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Basic Equation Solving

Definition

Basic equation solving involves finding values of variables that make an equation true. This fundamental algebraic skill requires understanding equality, inverse operations, and the properties of equations.

Prerequisites

To understand basic equation solving, you should have mastered:

  1. Variables & Expressions
  2. Order of Operations
  3. Fractions & Decimals

Learning Objectives

After mastering basic equation solving, you should be able to:

  1. Understand the concept of equality and balanced equations
  2. Solve one-step equations using inverse operations
  3. Solve two-step equations
  4. Check solutions by substitution
  5. Translate word problems into equations
  6. Apply the properties of equality
  7. Identify equations with no solution or infinite solutions

Key Concepts

Equation Properties

  • Addition property of equality
  • Subtraction property of equality
  • Multiplication property of equality
  • Division property of equality

Types of Equations

  • One-step equations
  • Two-step equations
  • Equations with variables on both sides
  • Equations with special solutions (no solution, all solutions)

Solution Methods

  • Inverse operations
  • Balance method
  • Substitution
  • Check solutions

Examples

Example 1: One-Step Equations (Grade 6)

Problem: Solve x + 5 = 12

Solution: 1. Subtract 5 from both sides x + 5 - 5 = 12 - 5 2. Simplify x = 7 3. Check: 7 + 5 = 12 ✓

Example 2: Two-Step Equations (Grade 7)

Problem: Solve 2x - 3 = 9

Solution: 1. Add 3 to both sides 2x - 3 + 3 = 9 + 3 2x = 12 2. Divide both sides by 2 2x ÷ 2 = 12 ÷ 2 x = 6 3. Check: 2(6) - 3 = 9 ✓

Example 3: Word Problem (Grade 7)

Problem: A number plus twice itself equals 24. What is the number?

Solution: 1. Let x = the number 2. Write equation: x + 2x = 24 3. Combine like terms: 3x = 24 4. Divide both sides by 3: x = 8 5. Check: 8 + 2(8) = 24 ✓

Common Misconceptions

  1. Not performing the same operation on both sides
  2. Forgetting to check solutions
  3. Confusing addition and multiplication properties
  4. Not recognizing special cases (no solution, all solutions)

Progression Path

This skill leads to:

  1. Linear Equations
  2. Linear Inequalities
  3. Systems of Equations
  4. Quadratic Equations

Practice Activities

  1. Basic: Solve one-step equations
  2. Intermediate: Solve two-step equations
  3. Advanced: Solve equations with variables on both sides
  4. Challenge: Create and solve word problems