Direct Variation
Direct variation happens when two variables change together in the same direction. If one increases, the other increases at the same rate.
Fundamental Principles
Direct Variation (Y ∝ X)
A relationship where the ratio of two variables is always equal to a constant value. This is written as y = k * x, where 'k' is the constant of variation.
Essential Formulation Tips
- When two values change by direct variation, you can set them up as an equal ratio: x1 / y1 = x2 / y2.
- Common real-world examples include purchasing items (more items cost more money) or working at a fixed hourly wage.
Shortcut Execution Techniques
- If a problem states that y varies directly with x, find the multiplier value 'k' using your initial data points to quickly solve for any new values.
Contextual Inquiries (FAQs)
Q: Does direct variation always draw a straight line on a graph?
A: Yes, it creates a straight line that passes directly through the origin (0, 0).
Example Breakdown: Solving Direct Resource Scaling
Foundational linear scaling problem.Set up the direct variation ratio: x1 / y1 = x2 / y2 -> 15 / 450 = 25 / y2.
Simplify the first ratio: 1 / 30 = 25 / y2.
Cross-multiply to solve for y2: y2 = 25 * 30.
Calculate the final value: y2 = $750.
Direct Variation Problems
Practice scaling directly proportional values and resource costs.
Q1. If a vehicle travels 120 miles on 4 gallons of fuel, how many miles can it travel on 6 gallons under identical conditions?