What does this calculator do?

This calculator works with the inverse variation relationship y = k/x. The same relationship can be written as x · y = k, which means the product of x and y stays constant.

• It finds k from one known point: k = x · y.
• It solves y when k and x are known: y = k / x.
• It solves x when k and y are known: x = k / y.
• It checks two points by comparing k1 = x1·y1 with k2 = x2·y2.

What is inverse variation?

Inverse variation means that two variables change in a way that keeps their product constant. In school math, it is also called inverse proportion.

A useful mental model is this: as x increases, y often decreases so that x · y remains the same k value. This model is common in algebra, proportion problems, and function interpretation.

How does the formula work?

The core formula is y = k/x. The constant k is found by multiplying x and y, and every valid point on the same inverse variation curve has the same k.

• k = x · y
• y = k / x
• x = k / y
• Two-point check: if x1·y1 = x2·y2, the points share the same inverse variation.

The calculator is numeric. It does not parse symbolic equations or solve a full algebra expression typed as text.

Inverse variation vs direct variation

Direct variation keeps a ratio constant; inverse variation keeps a product constant. This is the cleanest way to separate the two ideas.

• Direct variation: y = kx and k = y/x.
• Inverse variation: y = k/x and k = x·y.
• In direct variation, doubling x can double y.
• In inverse variation, doubling x usually halves y when k stays fixed.

For example, if x = 3 and y = 8, inverse variation gives k = 24. With the same k, x = 6 gives y = 24/6 = 4. x increased, y decreased, and the product stayed 24.

How to find k from one point

The Find k mode calculates k = x · y from a known point (x, y). Once k is known, it defines the inverse variation equation for that point.

1. Enter x = 4 and y = 6.
2. Compute k = x · y = 4 × 6 = 24.
3. The relationship is y = 24/x.
4. If x later becomes 8, y = 24/8 = 3.

How to solve for a missing x or y

The Solve mode uses division once k and one variable are known. You choose whether the missing target is y or x.

1. Solving y: if k = 36 and x = 9, then y = 36 / 9 = 4.
2. Solving x: if k = -20 and y = 5, then x = -20 / 5 = -4.
3. The calculator highlights the target result and shows the substitution row.

How to check two points

The Verify mode checks whether two points belong to the same inverse variation by calculating each point's k value separately.

1. First point: (3, 8), so k1 = 3 × 8 = 24.
2. Second point: (6, 4), so k2 = 6 × 4 = 24.
3. Because k1 and k2 are equal, both points fit y = 24/x.

If the second point were (4, 5), then k2 = 20. Since 24 and 20 are not equal, the two points would not describe the same inverse variation.

Why zero matters in inverse variation

Zero is sensitive in inverse variation because y = k/x is undefined when x = 0. A point with x = 0 cannot be a normal point on an inverse variation curve.

• Solve y cannot use x = 0.
• Solve x cannot use y = 0.
• Verify cannot accept x1 = 0 or x2 = 0 as valid inverse-variation points.
• k = 0 can appear arithmetically, but it is not a typical inverse variation constant.

How to read the visual graph

The graph is a lightweight orientation visual for the inverse variation curve. It helps you see the general hyperbola shape, but it is not a precision graphing tool.

When k is positive, the curve typically appears in quadrants I and III. When k is negative, it appears in quadrants II and IV. The plotted points help connect the calculation to the curve.

How to use the calculator

Use the mode that matches the value you are trying to find. Each mode has its own required fields.

1. Choose Find k if you know one point (x, y).
2. Choose Solve and target y if you know k and x.
3. Choose Solve and target x if you know k and y.
4. Choose Verify if you want to compare two points.
5. Use the sign control when a negative value is needed.

Decimal comma and decimal dot input can both be accepted. Ambiguous or invalid input should not be silently turned into another number.

Common mistakes

The most common mistake is using y/x as the constant. That belongs to direct variation, not inverse variation.

• Confusing y = kx with y = k/x.
• Using x = 0 as if it were a valid point.
• Treating k = 0 as a normal inverse variation relationship.
• Comparing ratios instead of products when checking two points.
• Reading the schematic graph as a precise graphing utility.

Limitations of this calculator

This calculator is a numeric inverse variation helper. It is not a full algebra solver, table generator, or precision graphing application.

• It does not solve direct variation y = kx.
• It does not handle joint or combined variation.
• It does not accept symbolic equation strings.
• It does not generate a full table of values.
• The graph is illustrative rather than a full graphing tool.
• It does not decide whether a real-life word problem truly fits inverse variation.

If you need y = kx, use the direct variation calculator. If you need a general proportion setup or cross multiplication, the ratio/proportion and cross multiplication calculators are more appropriate.