Math
Direct Variation Calculator
If y varies directly with x, then y = kx for a fixed constant k.
Give the direct variation calculator one point (x₁, y₁) and it finds k = y₁/x₁ as an exact fraction, writes the equation, and then finds y for any new x or x for any new y. The graph shows the straight line through the origin with slope k.
Part of a point (x₁, y₁) that satisfies the variation. Cannot be 0.
The y value that goes with x₁.
The x value you want the matching y for.
Try an example
Result
Answer
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- Constant of variation k
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- Equation
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- Missing value
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- Second point
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- Ratio check
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Student quick launch
Grade planning, algebra checks, and formulas students reach for most.
Study path
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Follow these when you want the formula behind the answer, a short lesson, or nearby tools in the same topic.
What the direct variation calculator solves
Direct variation (direct proportion) means two quantities keep the same ratio: y/x is always the constant of variation k. Word problems say 'y varies directly as x' or 'y is directly proportional to x'. Once one pair of values is known, k is fixed and every other pair follows.
Example: y = 10 when x = 4
- k = 10/4 = 5/2.
- Equation: y = (5/2)x.
- When x = 7: y = (5/2)(7) = 35/2 = 17.5.
- When y = 25: x = 25 ÷ (5/2) = 10.
| Type | Equation | Graph |
|---|---|---|
| Direct variation | y = kx | Line through the origin |
| Inverse variation | y = k/x | Hyperbola |
| Joint variation | z = kxy | z grows with both x and y |
How to use it
- Choose whether you want y for a new x, x for a new y, or only k and the equation.
- Enter the known point (x₁, y₁) from the problem.
- Enter the new x or y and read the answer, the constant k, and the equation. The ratio check confirms both points have the same y/x.
How to read the answer
k is the How steep a line is: how much y changes for each step right in x. of the line and also the unit rate: how much y changes for each 1 unit of x. A positive k means y grows with x; a negative k means y falls as x grows. The missing value is shown as an exact fraction with its decimal.
Common mistakes and edge cases
- Using x = 0 for the known point: k = y/0 is undefined. Choose a point with x ≠ 0.
- Dividing the wrong way: k = y/x, not x/y.
- Treating any linear relationship as direct variation. There must be no constant term.
- If k = 0, y is always 0 and no y other than 0 is reachable.
Worked examples
Find y for a new x
y = 10 when x = 4; find y when x = 7 → 35/2
Answer
35/2 (17.5)
Find x for a new y
y = 10 when x = 4; find x when y = 25 → 10
Answer
10
Negative constant
y = -6 when x = 2 gives y = -3x
Answer
y = -3x
Decimal inputs stay exact
y = 1.5 when x = 0.5 gives k = 3; at x = 2.2, y = 33/5
Answer
33/5 (6.6)
Zero constant
y = 0 when x = 3 gives k = 0, so y = 0 everywhere
Answer
0
x₁ = 0 is invalid
k = y/0 is undefined
Answer
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Frequently asked questions
What is the constant of variation?+
The fixed ratio k = y/x in y = kx. It is also the slope of the line and the unit rate of change.
How do I know a table shows direct variation?+
Divide y by x in every row. If every quotient is the same number k (and the table includes or would pass through (0, 0)), it is direct variation.
What does the graph of a direct variation look like?+
A straight line through the origin. Positive k rises to the right; negative k falls to the right.
Is direct variation the same as proportion?+
Yes. 'y is directly proportional to x' and 'y varies directly as x' both mean y = kx. Solving y₁/x₁ = y₂/x₂ by cross-multiplying gives the same answers.
Can k be negative or a fraction?+
Yes. k can be any non-zero real number. For example, y = -6 when x = 2 gives k = -3, and y = 10 when x = 4 gives k = 5/2.
What if y varies directly with x squared?+
Then y = kx^2, and k = y/x^2. Find k from the known point the same way, then substitute. This calculator handles the linear case y = kx; use the same steps by hand for powers.
About this calculator
- Written by
- mathcheck editorial team
- Last reviewed
- September 4, 2026
Method
- Uses the values entered by the user and stable formulas documented on the page.
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Last updated: September 4, 2026