1. What is Variation?

Variation describes how one quantity changes in relation to another quantity. In mathematics, we often study two main types of variation: direct (constant) variation and inverse variation.

2. Types of Variation

Direct Variation (y = kx): When two quantities change in the same direction. If x increases, y increases proportionally.

Inverse Variation (y = k/x): When two quantities change in opposite directions. If x increases, y decreases proportionally.

In both cases, k is the constant of variation that relates the two variables.

3. How to Use This Calculator

Steps:

1. Select whether you're working with direct or inverse variation
2. Enter a known x value
3. Enter the corresponding y value
4. Click Calculate to find the constant of variation (k) and the complete equation

4. Practical Applications

Direct Variation Examples:

• Distance traveled at constant speed (distance = speed × time)
• Total cost of items (cost = price per unit × number of units)

Inverse Variation Examples:

• Brightness of light vs distance from source
• Time to complete a task vs number of workers

5. Frequently Asked Questions (FAQ)

Q1: How do I know if a relationship is direct or inverse variation?
A: If both variables increase or decrease together, it's direct. If one increases while the other decreases, it's inverse.

Q2: Can k be negative in variation equations?
A: Yes, k can be negative, which indicates an inverse relationship in direct variation or vice versa.

Q3: What if x is zero in inverse variation?
A: The equation becomes undefined (division by zero), so x cannot be zero in inverse variation.

Q4: How is this different from linear equations?
A: Direct variation is a special case of linear equations (y = mx + b where b = 0). Inverse variation is non-linear.

Q5: Can this calculator solve for x or y if k is known?
A: Currently it calculates k from x and y, but you could rearrange the equation to solve for other variables.