Inverse Variation Calculator – Guide & Formulas
Calculate inverse variation between two variables. Find the constant of proportionality k, predict y values, and solve y = k/x problems instantly with step-by-step solutions.
Calculate inverse variation instantly with our free calculator. Find the constant k from a known x,y pair and predict y for any new x using the formula y = k/x.
Key Takeaway
Use the free Inverse Variation Calculator to calculate inverse variation between two variables. find the constant of proportionality k, predict y values, and solve y = k/x problems instantly with step-by-step solutions. Get instant results with step-by-step explanations.
How to Use the Inverse Variation Calculator
- Enter a known x value from the inverse variation relationship.
- Enter the corresponding known y value.
- The calculator finds k = x × y automatically.
- Enter a new x value to predict the corresponding y using y = k/x.
The Formula
Variable Definitions
- y: The dependent variable that varies inversely with x
- x: The independent variable (must be non-zero)
- k: The constant of proportionality: k = x × y
- ∝: Symbol for "varies inversely as": y ∝ 1/x means y = k/x
Inverse Variation: y = 6 when x = 4
Find k and predict y when x = 8.
- Step 1: Given y = 6 when x = 4, compute k = x × y = 4 × 6 = 24.
- Step 2: The inverse variation equation is y = 24/x.
- Step 3: To predict y when x = 8: y = 24 / 8 = 3.
- Step 4: Verify product: 4 × 6 = 24 and 8 × 3 = 24. The product is constant.
Frequently Asked Questions
What is inverse variation?
Inverse variation is a relationship where one variable increases as the other decreases, maintaining a constant product: xy = k. Expressed as y = k/x. As x doubles, y halves.
How do I find the constant of variation k?
Multiply any known x value by its corresponding y value: k = x × y. This product is the same for every pair in an inverse variation relationship.
What does the graph of inverse variation look like?
The graph is a hyperbola — two curves approaching but never touching the x-axis and y-axis. For positive k, curves are in quadrants I and III. For negative k, in quadrants II and IV.
Can k be negative?
Yes. A negative k means y and x have opposite signs. The curves appear in quadrants II and IV. For example, y = -12/x has curves in the second and fourth quadrants.
What are real-life examples of inverse variation?
Examples include: speed vs. time for fixed distance (t = d/v), pressure vs. volume (Boyle's law: P = k/V), resistance vs. wire cross-section, and workers vs. time to complete a task.
How is inverse variation different from direct variation?
Direct variation: y = kx (ratio y/x is constant). Inverse variation: y = k/x (product xy is constant). In direct variation, both variables increase together. In inverse, one increases while the other decreases.
Can x be zero in inverse variation?
No. Since y = k/x, x cannot be zero (division by zero is undefined). The y-axis is a vertical asymptote of the hyperbola.
What if I have multiple data points?
For inverse variation, k should be the same for all pairs. Compute k = x × y for each pair. If they are all equal (or very close), the relationship is truly inverse variation.