Direct Variation Calculator
Calculate the constant of variation and solve direct variation equations (y = kx)
Direct Variation: y = kx
Find the constant of variation (k) or solve for x or y given k
Enter x and y values to find the constant of variation (k = y/x)
Independent variable (cannot be 0)
Dependent variable
Enter k and one variable to find the other using y = kx
Quick Examples
Constant of Variation
k =
y = x
Solved for y
y =
When k = and x =
Solved for x
x =
When k = and y =
Step-by-Step Solution
Direct Variation Reference
| To Find | Formula | Need |
|---|---|---|
| k | k = y ÷ x | x and y values |
| y | y = k × x | k and x values |
| x | x = y ÷ k | k and y values |
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About Direct Variation Calculator
What is Direct Variation?
Direct variation describes a relationship between two variables where one is a constant multiple of the other. When two quantities vary directly, they increase or decrease proportionally.
The Direct Variation Formula
y = kx
Where:
- y is the dependent variable
- x is the independent variable
- k is the constant of variation (or constant of proportionality)
Finding the Constant of Variation
To find k, rearrange the formula:
k = y ÷ x
The constant k tells you how much y changes for every unit change in x.
Key Properties
- Passes through origin: A direct variation equation always passes through the point (0, 0)
- Constant ratio: The ratio y/x is always equal to k for any point on the graph
- Straight line: The graph is always a straight line through the origin
- Slope equals k: The constant of variation k is the slope of the line
How to Recognize Direct Variation
A relationship shows direct variation if:
- When x doubles, y doubles
- When x triples, y triples
- The ratio y/x is constant for all values
- The graph passes through the origin
Examples in Real Life
Distance and Time
If you drive at a constant speed of 60 mph:
- y = 60x (distance = 60 × time)
- k = 60 mph
Cost and Quantity
If apples cost $2 each:
- y = 2x (total cost = 2 × number of apples)
- k = 2 dollars per apple
Hourly Wages
If you earn $15 per hour:
- y = 15x (earnings = 15 × hours worked)
- k = 15 dollars per hour
Common Mistakes to Avoid
- Not checking for origin: If the equation doesn't pass through (0, 0), it's not direct variation
- Confusing with linear equations: y = mx + b is linear but only direct variation when b = 0
- Negative k values: k can be negative, representing inverse relationships
Frequently Asked Questions
Can k be zero?
No, if k = 0, then y would always equal 0 regardless of x, which isn't a useful variation relationship.
Can k be negative?
Yes! A negative k means that as x increases, y decreases (and vice versa). The variables still vary directly, just in opposite directions.
What's the difference between direct and inverse variation?
- Direct variation: y = kx (y increases as x increases)
- Inverse variation: y = k/x (y decreases as x increases)