Direct variation
Use y = kx. Multiplying x by a factor multiplies y by the same factor. The graph is a straight line through the origin.
k = y/x
y₂ = 28
Both pairs have the same direct-variation constant.
Direct and Inverse Variation Calculator
Choose the relationship, enter two corresponding x–y pairs, and get the missing value, constant of variation k, invariant check, and worked steps.
Enter two corresponding pairs
Variation result
Missing value
Relationshipy = 4x
Constant k4
Completed pairs(3, 12) and (7, 28)
Invarianty₁/x₁ = y₂/x₂ = 4
Verification12/3 = 28/7 ✓
Advertisement
Step-by-step solution
Advertisement
Direct vs. inverse variation
Inverse variation
Use y = k/x, equivalently xy = k. Multiplying x by a nonzero factor divides y by that factor.
k = xy
Quick comparison
| Feature | Direct | Inverse |
|---|---|---|
| Equation | y = kx | y = k/x |
| Constant check | y/x = k | xy = k |
| Two-pair equation | y₁x₂ = y₂x₁ | x₁y₁ = x₂y₂ |
| Zero values | (0,0) lies on every direct-variation graph | x = 0 is undefined; standard inverse variation uses k ≠ 0 |
How to solve variation problems
1. Identify the pattern
Choose direct when the ratio y/x stays fixed. Choose inverse when the product xy stays fixed.
2. Find k
Use one complete pair. Divide for direct variation; multiply for inverse variation.
3. Substitute
Put the known value from the other pair into y=kx or y=k/x.
4. Verify both pairs
Recalculate the invariant with both pairs. Matching values confirm the relationship, allowing for displayed rounding.
Units of the constant
For direct variation, the units of k are y-units per x-unit, such as miles per hour. For inverse variation, they are the product of the two units, such as worker-hours. Keep corresponding units consistent between Pair 1 and Pair 2.
Accuracy, assumptions, and privacy
The calculator evaluates decimal and fractional inputs with JavaScript floating-point arithmetic, displays up to 12 significant digits, and verifies completed pairs with a relative tolerance of 1 × 10−10. Values whose intermediate products overflow the supported range are rejected instead of showing infinity.
Standard inverse variation here means y=k/x with k≠0, so neither coordinate may be zero. Direct variation permits the origin, but at least one pair with nonzero x is needed to determine k. All calculations run locally in your browser; inputs are not uploaded or stored by this tool. Last reviewed: 4 August 2026.
Direct and inverse variation FAQs
What is direct variation?
Direct variation follows y=kx. Its ratio y/x is constant wherever x≠0, and its graph passes through the origin.
What is inverse variation?
Inverse variation follows y=k/x, or xy=k, with a nonzero constant k. As one variable grows in magnitude, the other shrinks proportionally.
How do I find the constant of variation?
For direct variation, calculate k=y/x from a pair with nonzero x. For inverse variation, calculate k=xy.
How do I know which variation type to use?
Test several pairs from the relationship. A constant quotient y/x signals direct variation; a constant product xy signals inverse variation.
Can direct variation include zero?
Yes. Because y=kx, setting x=0 also gives y=0. The origin alone cannot determine k, so another nonzero-x pair is required.
Why can’t inverse variation use zero?
The expression k/x is undefined at x=0. Under the standard assumption k≠0, y cannot be zero either.
Can I check completed pairs instead of solving?
Yes. Fill all four fields. The calculator compares the two ratios for direct variation or the two products for inverse variation and reports whether they match.