Direct Variation Calculator

Enter two known values to solve for the third in a direct variation relationship. Provide any two of x, y, or the constant of variation (k) — and this calculator solves the missing value using the formula y = k · x. You can also choose from variation types like y varies directly as x, y varies directly as the square of x, or y varies directly as the cube of x to cover a wider range of proportionality problems. Also try the Radical Simplifier.

The proportionality constant. Leave blank if solving for k.

Leave blank if solving for x.

Leave blank if solving for y.

Results

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Formula Used

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Constant of Variation (k)

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x Value

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y Value

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Results Table

Frequently Asked Questions

What is direct variation?

Direct variation (or direct proportionality) describes a relationship between two variables where an increase in one causes a proportional increase in the other. Mathematically, it is written as y = kx, where k is the constant of variation. The graph of a direct variation always passes through the origin. See also our calculate Cubic Equation.

How do you find the constant of variation (k)?

To find k, rearrange the direct variation formula: k = y / x (for linear variation). Simply divide the known y value by the corresponding x value. For example, if y = 12 and x = 4, then k = 12 / 4 = 3.

How do you recognize direct variation?

A relationship is a direct variation if the ratio y/x (or y/f(x) for other types) is constant for all data points. On a graph, a direct variation always produces a straight line passing through the origin (0, 0). If the line does not pass through the origin, it is not a direct variation.

What is y in the direct variation y = 3x, at x = 8?

Using the formula y = kx, substitute k = 3 and x = 8: y = 3 × 8 = 24. So y equals 24 when x is 8 in a direct variation with constant k = 3. You might also find our Reverse FOIL useful.

What is the difference between direct variation and inverse variation?

In direct variation (y = kx), when x increases, y increases proportionally. In inverse variation (y = k/x), when x increases, y decreases. Direct variation graphs are straight lines through the origin, while inverse variation graphs are hyperbolas.

What does 'y varies directly as the square of x' mean?

This means y = kx², so y is proportional to x squared rather than x itself. Doubling x will quadruple y. This type of variation is common in physics — for example, the kinetic energy of an object varies directly as the square of its velocity.

What is a real-life example of direct variation?

A classic example is the relationship between distance and time at constant speed: distance = speed × time. Here, speed acts as the constant of variation k. Other examples include the cost of items (total cost = price per item × quantity) and gravitational force proportional to mass.

Can the constant of variation k be negative?

Yes, k can be negative. A negative k means the dependent variable y decreases as x increases, but the rate of change is still constant. For example, y = -2x means for every 1 unit increase in x, y decreases by 2 units.