Lecture Notes: First Degree Equations of One Variable

Introduction to First Degree Equations

• Definition:
  • An equation is a statement that two mathematical expressions are equal.
• Solutions:
  • Values that make the equation a true statement.

Characteristics of First Degree Equations

• Involves only constants or variables with an exponent of 1.
• Example:
  • Equation: 3x - 1 = x + 7
  • Solution: x = 4 (Verified by substituting x with 4)

Solving First Degree Equations

1. Properties of Equality:
   • Add/Subtract: a + c = b + c and a - c = b - c
   • Multiply/Divide: a * c = b * c and a / c = b / c (c ≠ 0)
2. Example Problem:
   • Solve: 6k - (2k - 2) = 2(k - 1) + 10
   • Steps:
     • Simplify both sides
     • Distribute minus sign and 2
     • Combine like terms
     • Apply properties of equality (Add/Subtract/Divide)
     • Solution: k = 3

Solving Equations with Fractions

• Example: Solve 10 / (3n + 3) = 4 - 2 / (n + 1)
  • Eliminate fractions using Least Common Denominator (LCD)
  • Multiply both sides by LCD
  • Simplify and solve
  • Solution: n = 1/3

Absolute Value Equations

• Definition:
  • Non-negative distance from 0
  • Absolute value |y| = c implies y = c or y = -c
• Example: Solve |4x + 3| = 9
  • Set up two equations: 4x + 3 = 9 and -(4x + 3) = 9
  • Solve both for x
  • Solutions: x = -3 and x = 3/2

Solving Applied Problems with Equations

1. Steps to Solve Applied Problems:
   • Read the problem carefully
   • Identify unknowns
   • Choose variable(s) for unknowns
   • Create sketches or charts if necessary
   • Write the equation
   • Solve the equation
   • Check the solution in context
2. Example Problem:
   • $20,000 invested at 4%, find additional amount at 5% for 4.2% yield
   • Define unknown (additional amount as x)
   • Set up equation with interest expressions
   • Solve: x = 5,000
   • Validate solution with calculations and contextual reasoning

Conclusion

• Emphasizes importance of understanding context and verifying solutions both mathematically and contextually.