Inverse variation is a relationship between two variables where one goes up exactly as the other goes down so that their product stays constant.

Quick Scoop: What Is Inverse Variation?

  • Two quantities xxx and yyy are in inverse variation if

xy=kxy=kxy=k

where kkk is a non-zero constant.

  • Equivalently, you can write

y=kxy=\frac{k}{x}y=xk​

which shows that yyy is proportional to the reciprocal of xxx.

  • As xxx increases, yyy decreases in such a way that their product does not change, and vice versa.

If doubling one quantity makes the other cut in half (so the product is unchanged), you’re looking at an inverse variation.

Simple Real-Life Examples

  • Speed and travel time (fixed distance) : If you drive faster, the time needed to cover the same distance goes down; distance × time is constant for a fixed speed pattern.
  • Pressure and volume of a gas (at constant temperature) : In basic physics, increasing pressure often decreases volume so that the product PVPVPV is (approximately) constant (Boyle-type behavior).

These are classic “one up, one down” situations where the product stays the same.

Quick Formula Facts

  • Definition: xy=kxy=kxy=k (constant of variation k≠0k\neq 0k=0).
  • To find kkk, use any known pair: k=x1y1k=x_1y_1k=x1​y1​.
  • To find a missing value:
    1. Use a known pair (x1,y1)(x_1,y_1)(x1​,y1​) to get k=x1y1k=x_1y_1k=x1​y1​.
    2. Use xy=kxy=kxy=k to solve for the unknown xxx or yyy.

Example: If yyy varies inversely as xxx, and y=8y=8y=8 when x=3x=3x=3:

  • k=3⋅8=24k=3\cdot 8=24k=3⋅8=24.
  • Equation: y=24xy=\dfrac{24}{x}y=x24​.
  • If x=10x=10x=10, then y=2410=2.4y=\dfrac{24}{10}=2.4y=1024​=2.4.

How It Looks on a Graph

  • The graph of an inverse variation y=kxy=\dfrac{k}{x}y=xk​ is a rectangular hyperbola.
  • It never touches the x-axis or y-axis because xxx and yyy cannot be zero.

This curved, “bending” shape is what distinguishes inverse variation from straight-line (direct) variation.

Direct vs Inverse at a Glance

Feature Direct Variation Inverse Variation
Basic formula $$y = kx$$ $$y = \dfrac{k}{x}$$
What stays constant? $$\dfrac{y}{x} = k$$ $$xy = k$$
How variables move Both increase or both decrease together One increases while the other decreases
Graph shape Straight line through the origin Rectangular hyperbola
[7][3][5] **TL;DR:** Inverse variation means xxx and yyy are linked by xy=kxy=kxy=k, so if one goes up, the other goes down in just the right way to keep their product constant.

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