A direct variation equation describes a relationship where two variables increase or decrease proportionally. This means if one variable doubles, the other variable also doubles.
The general form of a direct variation equation is:
y = kx
Where:
- y and x are the variables
- k is the constant of variation
Understanding the Equation:
- k represents the factor by which y changes for every unit change in x.
- If k is positive, y increases as x increases.
- If k is negative, y decreases as x increases.
How to Find the Equation:
- Identify the variables: Determine which two variables are related by direct variation.
- Find the constant of variation (k): Use a given pair of values for the variables (x, y) to solve for k in the equation y = kx.
- Write the equation: Substitute the value of k back into the general equation y = kx.
Example:
Suppose the distance traveled (d) varies directly with the time (t) spent traveling. If you travel 60 miles in 2 hours, what is the direct variation equation relating distance and time?
- Variables: Distance (d) and time (t)
- Constant of variation (k): We know d = 60 miles when t = 2 hours. Substitute these values into the equation d = kt:
- 60 = k * 2
- k = 60 / 2 = 30
- Equation: Substitute k = 30 back into the equation d = kt:
- d = 30t
This equation tells us that for every hour traveled, the distance traveled increases by 30 miles.
Practical Insights:
- Direct variation is common in real-world scenarios: Think about the relationship between the number of hours worked and the amount of money earned, or the number of items purchased and the total cost.
- Understanding direct variation can help you make predictions: Once you know the equation, you can predict the value of one variable if you know the value of the other.
