Hong Life CalculatorsUnderstanding the direct variation calculator — concept and principle
This calculator works with direct variation y = k·xⁿ. From one known pair it finds the constant of variation k, writes the equation, and gives the missing y for a new x or the missing x for a new y. Use it for algebra word problems, scaling recipes, or physics relations such as distance at constant speed.
How it is calculated
k = y₁ ÷ x₁ⁿ
New y = k × xⁿ; new x = (y ÷ k)^(1/n)
Factor of change in y = new y ÷ y₁
- n = 1 is plain direct variation; n = 2 means y varies with the square of x.
- Check: y ÷ xⁿ stays equal to k for every pair.
The principle
In a direct variation, y is always the same multiple of xⁿ. One known pair is enough to find that multiple, and then every other pair follows. With n = 1, doubling x doubles y; with n = 2, doubling x multiplies y by 4. Examples are distance at a constant speed (n = 1), the area of a circle (n = 2, k = π) and the distance fallen from rest (n = 2, k ≈ 4.9 m/s²).
Worked example
Default inputs: x₁ = 4, y₁ = 10, n = 1, find y for x = 7.
- Constant of variation k = 10 ÷ 4 = 2.5.
- Equation: y = 2.5x; known pair (4, 10).
- New y = 2.5 × 7 = 17.5 (what was found: y for the new x).
- Factor of change in y = 17.5 ÷ 10 = 1.75, the same as 7 ÷ 4.
- Check: y ÷ x = 17.5 ÷ 7 = 2.5 = k.
| Known x (x₁) | Answer |
|---|---|
| 1 | 70 |
| 2 | 35 |
| 3 | 23.33333333 |
| 4 | 17.5 |
| 5 | 14 |
| 6 | 11.66666667 |
| 8 | 8.75 |
How to read the result
k tells you how much y there is per unit of xⁿ, for example 2.5 euros per kilogram. With n = 1, the factor of change in y equals the factor of change in x; with n = 2 it is the square of that factor. If your data do not keep y ÷ xⁿ constant, the relationship is not a direct variation of that power.
Common mistakes
- Computing k as x ÷ y.
- Forgetting the power n, for example when area varies with the square of length.
- Using direct variation when there is a fixed starting amount (y = kx + b is not direct variation).
- Taking an even root of a negative number when solving for x.
Frequently asked questions
What is direct variation?
A relationship y = kxⁿ where y is always the same multiple of xⁿ.
How do I find the constant of variation?
Divide y by xⁿ using any known pair.
What is the difference between direct and inverse variation?
In direct variation y grows with x; in inverse variation y = k ÷ x, so y shrinks as x grows.
Sources: OpenStax, College Algebra 2e (2021), §5.8 Modeling Using Variation