More Linear Equations: Lecture Notes

Introduction

• Focus on three examples of linear equations.
• Highlight some unique cases in solving linear equations.
• Emphasize recognizing these cases, which may seem complex but are often simpler.

Example 1: Solving for x where the result is zero

• Combine terms on one side of the equation.
• Simplify equation to 5x = 0.
• Solve by dividing both sides by 5, resulting in x = 0.
• Verify by substituting x = 0 back into the equation.
• Key takeaway: 0 is a legitimate solution.

Example 2: True Statement Leads to Infinite Solutions

• Start by distributing terms: 2x + 4 + 2x = 4x + 4.
• Combine terms to simplify both sides.
• After simplification, both sides equal, i.e., 4 = 4.
• Since this is a true statement, x can be any real number.
• Conclusion: Infinite solutions, x can be any value.

Example 3: False Statement Leads to No Solution

• Again, distribute terms as needed.
• Combine terms to isolate x.
• Resulting in a false statement: 20 ≠ 21.
• Conclusion: No solution exists.
• Mathematically represented as an empty set.

Conclusion

• Summary of different cases:
  • True Statement: Infinite solutions.
  • False Statement: No solution.
• Importance of following steps and recognizing the nature of the solution.

Note

• Encourage practicing more such examples to master recognizing these cases.