1. What is Direct and Inverse Variation?

Direct variation describes a relationship where y changes directly with x (y = kx), while inverse variation describes a relationship where y changes inversely with x (y = k/x). These are fundamental concepts in algebra and physics.

2. How Does the Calculator Work?

The calculator uses the following equations:

Where:

• \( y \) — Dependent variable
• \( x \) — Independent variable
• \( k \) — Constant of variation

Explanation: Given one pair of (x,y) values, the calculator finds k, then uses it to calculate a new y value for a given x.

3. Practical Applications

Examples: Direct variation appears in Hooke's Law (spring force), while inverse variation appears in Boyle's Law (gas pressure-volume relationship).

4. Using the Calculator

Steps:

1. Select variation type (direct or inverse)
2. Enter known x and y values
3. Enter new x value
4. Get the constant of variation, equation, and new y value

5. Frequently Asked Questions (FAQ)

Q1: How do I know if a relationship is direct or inverse?
A: In direct variation, y increases as x increases. In inverse variation, y decreases as x increases.

Q2: What if x is zero in inverse variation?
A: The calculator prevents division by zero. In reality, x cannot be zero in inverse variation.

Q3: Can k be negative?
A: Yes, k can be negative, resulting in negative variation relationships.

Q4: How is this different from linear equations?
A: Direct variation is a specific linear case (y = kx, no y-intercept). Inverse variation is non-linear.

Q5: What are real-world examples?
A: Direct: distance vs time at constant speed. Inverse: brightness vs distance from light source.