Lecture Notes: Constant of Proportionality

Understanding Constant of Proportionality

• Definition: Between two variables, y and x, the constant of proportionality is a constant value, k, such that:
  • ( y = kx )
• Expressing:
  • ( \frac{y}{x} = k )
  • This means for any point ((x, y)) on the line, the ratio ( \frac{y}{x} ) gives the constant of proportionality.

Example 1: Finding Constant of Proportionality

• Given Point: (3, 2)
• Equation:
  • ( y = kx )
  • Substitute: ( 2 = k \times 3 )
• Solution:
  • ( k = \frac{2}{3} )
  • Constant of proportionality is ( \frac{2}{3} )

Example 2: Identifying Line with Given Constant of Proportionality

• Goal: Find which line has a constant of proportionality ( \frac{5}{4} )
• Test Line A:
  • Point on A: (2, 5)
  • ( \frac{5}{2} \neq \frac{5}{4} )
  • Not the correct line.
• Test Line B:
  • Point on B: (4, 5)
  • ( \frac{5}{4} = \frac{5}{4} )
  • Correct line is B.

Special Case: When y = x

• Scenario:
  • ( y = x )
  • This can be rewritten as ( y = 1 \times x )
• Constant of Proportionality:
  • ( k = 1 )
  • Graph: Line at 45-degree with slope of 1
  • Represents the line y = x on a graph, indicating constant proportionality of 1.

Key Takeaways

• The constant of proportionality is the factor that relates two proportional variables.
• It can be identified by dividing y by x for any point on the line.
• Understanding the constant helps in predicting the behavior and relation between the variables.