Direct Variation vs. Inverse Variation - Algebra-2

1. Fundamental Concepts

Definition: Direct Variation: A relationship where one variable is a constant multiple of the other, expressed as $$y = kx$$ , where $$k$$ is a non-zero constant.
Definition: Inverse Variation: A relationship where the product of two variables is a constant, expressed as $$xy = k$$ or $$y = \frac{k}{x}$$ , where $$k$$ is a non-zero constant.
Graphical Representation: Direct variation forms a straight line through the origin, while inverse variation forms a hyperbola with asymptotes on the coordinate axes.

Direct Variation Rule: $$y = kx$$

Inverse Variation Rule: $$xy = k$$

Application: Used in physics (e.g., Hooke's Law for direct variation), economics (supply and demand for inverse variation)

Problem: Determine if the relationship between $$x$$ and $$y$$ is direct or inverse given $$y = \frac{5}{x}$$ .

Identify the form of the equation: $$y = \frac{5}{x}$$ matches the form of inverse variation $$y = \frac{k}{x}$$ .
Conclusion: The relationship is an inverse variation with $$k = 5$$ .

Validation: Substitute $$x = 1$$ → Original: $$y = \frac{5}{1} = 5$$ ; Simplified: $$y = 5$$ ✓

Problem: If $$y$$ varies directly with $$x$$ and $$y = 10$$ when $$x = 2$$ , find the value of $$y$$ when $$x = 5$$ .

Use the direct variation formula: $$y = kx$$ . Given $$y = 10$$ when $$x = 2$$ , solve for $$k$$ : $$10 = k \cdot 2$$ , so $$k = 5$$ .
Find $$y$$ when $$x = 5$$ : $$y = 5 \cdot 5 = 25$$ .

Validation: Substitute $$x = 5$$ → Original: $$y = 5 \cdot 5 = 25$$ ; Simplified: $$y = 25$$ ✓

Identification Strategy: Recognize the form of the equation to determine if it represents direct or inverse variation.
Substitution Method: Use given values to find the constant $$k$$ and then apply it to find unknown values.
Graphical Interpretation: Plot points to visually confirm the type of variation (straight line for direct, hyperbola for inverse).