Direct and inverse variation formula reference
| Task | Direct variation | Inverse variation |
|---|---|---|
| Solve y | y = kx | y = k/x |
| Solve x | x = y/k | x = k/y |
| Solve k | k = y/x | k = xy |
| Two pairs | y₁/x₁ = y₂/x₂, or y₁x₂ = y₂x₁ | x₁y₁ = x₂y₂ |
Terms
- Independent variable (x)
- The input or variable chosen freely.
- Dependent variable (y)
- The output whose value depends on x.
- Constant of variation / constant of proportionality (k)
- The fixed multiplier in direct variation or fixed product in inverse variation.
- Invariant
- A quantity that does not change:
y/xfor direct variation orxyfor inverse variation.
How to choose the model
| Test | Direct variation | Inverse variation |
|---|---|---|
| Number pattern | y/x is constant | xy is constant |
| Graph | Straight line through the origin | Reciprocal curve approaching both axes |
| If x doubles | y doubles | y halves |
| Equation | y = kx | y = k/x |
Direct variation is not every linear equation
A general linear equation is y = mx + b. It is direct variation only when b = 0, so the graph passes through (0,0) and its slope m is the constant k.
Common mistakes
- For direct variation, keep the order
y/xconsistent; usingx/ygives the reciprocal constant. - A decreasing relationship is not automatically inverse variation. Its products
xymust stay constant. - Convert corresponding quantities to consistent units before comparing them.
- Do not treat rounded observations as perfectly exact; use the reported tolerance note.
How to solve variation problems
1. Choose the task and model
Choose solve, predict, check, or identify. Use direct when y/x is fixed and inverse when xy is fixed.
2. Enter the known values
Enter only the fields shown. Fractions, decimals, signed values, and scientific notation are accepted.
3. Find the invariant
Divide y by x for direct variation or multiply x and y for inverse variation.
4. Solve and verify
Substitute k, solve the requested value, and check the same ratio or product.
Worked examples
Direct variation: find k and a new y
Problem: If y = 10 when x = 4, find k and y when x = 6.
- Substitute:
10 = k(4). - Simplify:
k = 10/4 = 5/2. - Equation:
y = (5/2)x. - New value:
y = (5/2)(6) = 15. - Verify:
10/4 = 15/6 = 5/2.
Inverse variation: find a missing x
Problem: If y = 8 when x = 3, find x when y = 6.
- Substitute:
k = xy = 3(8). - Simplify:
k = 24. - Equation:
y = 24/x. - Solve:
x = k/y = 24/6 = 4. - Verify:
3(8) = 4(6) = 24.
Exact fraction example
Problem: In direct variation, y = 5/6 when x = 1/3. Find y when x = 3/2.
- Substitute:
5/6 = k(1/3). - Simplify:
k = (5/6) ÷ (1/3) = 5/2. - Equation:
y = (5/2)x. - Solve:
y = (5/2)(3/2) = 15/4. - Verify:
(5/6)/(1/3) = (15/4)/(3/2) = 5/2.
Classify a table
Problem: Classify (2,12), (3,8), and (4,6).
- Ratios:
12/2 = 6,8/3,6/4 = 3/2; they are not constant. - Products:
2·12 = 24,3·8 = 24,4·6 = 24. - Conclusion: inverse variation.
- Equation:
y = 24/x. - Verify each row by confirming
xy = 24.
Practical interpretation of k
Hourly pay — direct
At £18 per hour, pay y varies directly with hours x. Here k = 18 £/hour: the constant is the pay earned per hour.
Constant speed — direct
Travelling 180 miles in 3 hours gives k = 60 miles/hour. Distance equals speed times time.
Worker-hours — inverse
If 4 workers need 15 hours, k = 60 worker-hours. With a fixed workload, workers multiplied by hours stays 60.
Fixed-distance travel — inverse
For a 300-mile trip, speed times time is k = 300 miles. At 75 mph, time is 4 hours.
Review, accuracy, and privacy
Authored and reviewed by: Starlight Tools Mathematics Review Team, experienced in exact-arithmetic calculator implementation and algebra-content quality assurance. No individual credentialed reviewer is claimed because no verified individual attribution is available for this page.
Review method: formulas were checked by algebraic substitution, exact-fraction cases and zero restrictions were tested, table classifications were verified from both invariants, and definitions were compared with OpenStax Elementary Algebra 2e, §8.9 and Mathematics LibreTexts, Direct and Inverse Variation. Reviewed .
Integer, decimal, fraction, and scientific-notation inputs are converted to reduced fractions and calculated exactly. Decimal approximations are display aids. For tables or pairs representing rounded observations, invariants within one part in one billion of one another are judged consistent; the result explicitly says when this rounding allowance, rather than exact equality, was needed. Inputs are limited to 40 digits.
Standard direct and inverse variation use k ≠ 0. Inverse variation also excludes x = 0 and y = 0. Direct variation may include (0,0), but that point alone cannot determine k. All calculations run locally; values are not uploaded or stored.
Direct and inverse variation FAQs
Can the constant of variation k be negative?
Yes. A negative k gives a decreasing direct-variation line or inverse-variation branches in quadrants II and IV. In standard variation, k must not be zero.
Is inverse variation the same as indirect proportion?
In many school texts, indirect variation, inverse variation, and inverse proportion all mean y = k/x. Check the stated formula because terminology can vary.
How is direct variation different from a linear function?
Direct variation is the special linear function y = kx with y-intercept zero. A general linear function y = mx + b is direct variation only when b = 0.
How do I solve when k is already known?
In “Solve x, y, or k” mode, select x or y as the unknown and enter k plus the other variable. The calculator uses y = kx or y = k/x.
How do I classify an x-y table?
Compare y/x down the table for direct variation and xy down the table for inverse variation. “Identify the relationship” calculates both columns and marks inconsistent rows.
How are rounded data values handled?
All entered values are converted to exact fractions. In pair and table checks, invariants that differ by no more than one part in one billion are treated as matching rounded observations, and the result says when tolerance was used.