Inverse Variation K is the Constant of Variation - Algebra-2

1. Fundamental Concepts

Definition: Inverse Variation describes a relationship between two variables where their product is always a constant, denoted as $$K$$ . Mathematically, if $$x$$ and $$y$$ are inversely proportional, then $$xy = K$$ .
Constant of Variation: The constant $$K$$ in the equation $$xy = K$$ represents the product of the two variables and remains unchanged regardless of the values of $$x$$ and $$y$$ .
Graphical Representation: The graph of an inverse variation is a hyperbola with asymptotes at the coordinate axes.

Basic Rule: $$xy = K$$

Degree Preservation: The product of the variables remains constant regardless of their individual values.

Application: Used to model relationships where one variable increases while the other decreases proportionally.

Problem: If $$x$$ and $$y$$ are inversely proportional and $$x = 4$$ when $$y = 6$$ , find the constant of variation $$K$$ .

Validation: Substitute $$x=4$$ and $$y=6$$ → Original: $$4 \cdot 6 = 24$$ ; Simplified: $$K = 24$$ ✓

Problem: Given that $$xy = 30$$ , find $$y$$ when $$x = 5$$ .

Validation: Substitute $$x=5$$ and $$y=6$$ → Original: $$5 \cdot 6 = 30$$ ; Simplified: $$K = 30$$ ✓

Visual Strategy: Use graphs to visualize the inverse relationship and identify key points.
Error-Proofing: Always check the consistency of the product $$xy = K$$ after solving for one variable.
Concept Reinforcement: Practice with various scenarios to understand how changes in one variable affect the other.