Calculate inverse variation (y = k/x), inverse square (y = k/x²), and inverse cube (y = k/x³). Find the constant k, predict new values, and compare with direct variation.
Inverse variation describes a relationship in which one quantity increases while another decreases proportionally, such that their product remains constant. The simplest form is y = k/x, where k is the constant of variation. If you double x, y is halved; if you triple x, y becomes one-third. This relationship appears throughout science and everyday life — Boyle's gas law (pressure × volume = constant), the relationship between speed and travel time for a fixed distance, and electrical resistance and current for a fixed voltage are all examples.
Beyond simple inverse variation, the inverse square law y = k/x² governs phenomena such as gravitational force, light intensity, and electromagnetic radiation, where the effect diminishes with the square of the distance. The inverse cube law y = k/x³ applies to tidal forces and certain fluid dynamics relationships.
To use inverse variation in problem solving, you first determine the constant k from a known pair (x₁, y₁) using k = x₁ⁿ · y₁ where n is the power. Then, for any new x₂, you compute y₂ = k / x₂ⁿ. The constant k encodes the specific relationship — different physical systems have different k values but follow the same mathematical structure.
This calculator supports all three inverse models (1/x, 1/x², 1/x³), computes the constant from your known data point, predicts new values, generates a values table over a customizable range, and visually compares inverse variation against direct variation so you can see the fundamentally different behavior of these two relationship types.
Inverse Variation Calculator helps you solve inverse variation problems quickly while keeping each step transparent. Instead of redoing long algebra by hand, you can enter Known x₁, Known y₁, New x₂ (predict y₂) once and immediately inspect Constant of Variation (k), Equation, Product Check (x₁y₁) to validate your work.
This is useful for homework checks, classroom examples, and practical what-if analysis. You keep the conceptual understanding while reducing arithmetic mistakes in multi-step calculations.
y = k / xⁿ, where k = y₁ · x₁ⁿ. For n=1: xy = k (constant product). For n=2: x²y = k. Prediction: y₂ = k / x₂ⁿ.
Result: Constant of Variation (k) shown by the calculator
Using the preset "y=12/x, x₂=4", the calculator evaluates the inverse variation setup, applies the selected algebra rules, and reports Constant of Variation (k) with supporting checks so you can verify each transformation.
This calculator takes Known x₁, Known y₁, New x₂ (predict y₂), Table range start and applies the relevant inverse variation relationships from your chosen method. It returns both final and intermediate values so you can audit the process instead of treating it as a black box.
Start with the primary output, then use Constant of Variation (k), Equation, Product Check (x₁y₁), % Change in x to confirm signs, magnitude, and internal consistency. If anything looks off, change one input and compare the updated outputs to isolate the issue quickly.
A strong workflow is manual solve first, calculator verify second. Repeating that loop improves speed and accuracy because you learn to spot common setup errors before they cost points on multi-step algebra problems.
Inverse variation is a relationship where one variable increases as the other decreases, such that their product (or x^n · y) remains constant. The equation is y = k/x for simple inverse variation.
Multiply the known y-value by the known x-value (raised to the appropriate power). For y = k/x, k = x₁ · y₁. For y = k/x², k = x₁² · y₁.
In direct variation (y = kx), both variables increase or decrease together. In inverse variation (y = k/x), when one increases the other decreases. Direct gives a straight line through the origin; inverse gives a hyperbola.
The inverse square law states that a quantity is inversely proportional to the square of the distance: y = k/x². Examples include gravitational force, light intensity, and sound intensity.
Yes. If x and y have opposite signs, k will be negative. The mathematical form y = k/x still holds, but the graph occupies different quadrants of the coordinate plane.
Speed and time (fixed distance), pressure and volume (Boyle's law), gravitational force and distance, number of workers and time to complete a project, current and resistance (fixed voltage) are all classic examples. Understanding this concept helps you apply the calculator correctly and interpret the results with confidence.