Last updated: July 31, 2026
Direct Variation Calculator
Creators
Dharmendra SinghReviewers
Creators
Dharmendra SinghReviewers
Creators
Dharmendra SinghReviewers
Creators
Dharmendra SinghReviewers
Formula
k = y1 / x1, then y2 = k × x2
Where:
- x_1=Known Independent Value
- y_1=Known Dependent Value
- k=Constant of Variation
- x_2=Target Independent Value
- y_2=Predicted Dependent Value
Worked Examples
Positive proportional relationship
Given (x1, y1) = (2, 6), predict y2 at x2 = 5.
- 1k = y1/x1 = 6/2 = 3
- 2y2 = k×x2 = 3×5
- 3y2 = 15
Negative constant
Given (x1, y1) = (-4, 8), find y2 when x2 = 10.
- 1k = 8/(-4) = -2
- 2y2 = -2×10
- 3y2 = -20
Decimal values
Use x1 = 1.5, y1 = 3.75, and x2 = 4.2.
- 1k = 3.75 / 1.5 = 2.5
- 2y2 = 2.5 × 4.2
- 3y2 = 10.5
Invalid known x
x1 = 0 cannot define a direct variation constant.
- 1k = y1/x1 requires division by x1
- 2x1 = 0 makes k undefined
- 3No valid direct-variation model
Introduction
The Direct Variation Calculator models relationships where one quantity changes in exact proportion to another. It first computes the constant ratio k = y/x, then uses y = kx to predict new values reliably.
What Is Direct Variation?
Direct variation means two variables maintain a constant ratio, so the graph is a straight line through the origin.
Equation form: y = kx
k is constant for all valid points
Line passes through (0, 0)
Scaling x scales y proportionally
Finding the Constant k
From any known nonzero x and its paired y, calculate k = y/x. This constant defines the full model.
Need one known pair (x1, y1)
x1 must not be zero
Positive k means same-sign movement
Negative k means opposite-sign movement
Predicting New Values
After deriving k, predict any target output using y2 = k×x2.
Compute k once from known data
Multiply k by new x2
Interpret sign and magnitude
Use rounded output for reporting
Graph Interpretation
Direct variation always plots as a line through the origin. The slope of that line is k.
- k > 0:
rising line
- k < 0:
falling line
- |k| larger:
steeper line
All points satisfy y/x = k
How to Use This Calculator
Enter a known point and your target x-value to get both the constant and predicted output.
Enter x1 and y1
Enter target x2
Read k and y2 outputs
Fix x1 if you receive divide-by-zero error
Real-World Applications
Direct variation appears in unit pricing, scaling recipes, dosage scaling, and many linear-rate systems.
Cost vs quantity at fixed unit price
Distance vs time at constant speed
Recipe ingredient scaling
Linear sensor calibration
Direct vs Inverse Variation
Direct variation grows together; inverse variation trades off. Choosing the right model matters.
| Type | Equation | Pattern |
|---|---|---|
| Direct | y = kx | y increases when x increases (for k > 0) |
| Inverse | y = k/x | y decreases when x increases |
| Direct Graph | Line through origin | Constant slope |
| Inverse Graph | Hyperbola | Nonlinear curve |
FAQs
What is direct variation in simple terms?
It means one variable is always a constant multiple of another, written as y = kx.
How do I calculate the constant of variation?
Use k = y1/x1 from any known pair where x1 is not zero.
Why can't x1 be zero?
Because k = y1/x1 would require division by zero, which is undefined.
Can k be negative?
Yes. A negative k means y has the opposite sign of x and the graph slopes downward.
Does the graph always pass through the origin?
Yes, for direct variation models of the form y = kx, the point (0,0) is always included.
Is direct variation the same as linear equation y = mx + b?
It is a special case where b = 0. All direct-variation equations are linear, but not all linear equations are direct variation.
Can I use decimals in this calculator?
Yes. Decimal inputs are supported and results are rounded to six decimal places.