What is the difference between inverse and direct variation?

In direct variation, y = kx, and both variables change in the same direction: if x doubles, y doubles. In inverse variation, y = \frac{k}{x}, and the variables change in opposite directions: if x doubles, y is cut in half. Both have a constant, but direct variation multiplies and inverse variation divides.

How do I find the constant k in an inverse variation?

Multiply the two known values together: k = x \cdot y. For example, if x = 5 and y = 8 satisfy the relationship, then k = 5 \cdot 8 = 40, and the equation is y = \frac{40}{x}.

Can x or y be negative in inverse variation?

Yes. Negative values work just like positive values in the formula y = \frac{k}{x}. If k is positive and x is negative, then y will be negative. If both x and y are negative, then k is positive. Just be careful with signs when doing arithmetic.

Why does the graph of y = \frac{k}{x} never touch the axes?

The graph never crosses the x-axis because y can never equal zero (unless k = 0, which is not inverse variation). Similarly, the graph never crosses the y-axis because x can never equal zero. The axes are called asymptotes—the curve approaches them but never reaches them.

Inverse Variation

Inverse variation describes when two quantities always multiply to give a constant value, with one variable increasing as the other decreases.

y = \frac{k}{x}

Solve a problem with Inverse Variation

Type the problem. The solver will use Inverse Variation where Inverse Variation is the right tool, and tell you when it is not.

What each symbol means

What Inverse Variation takes

Symbol	Meaning
$k$	The constant of proportionality, which equals the product $xy$ no matter what values $x$ and $y$ have; it must be non-zero for the inverse relationship to exist.
$x$	The independent variable, which can be any real number except zero (since division by zero is undefined).
$y$	The dependent variable, the output of the inverse variation formula; it equals zero only if $k = 0$ , which breaks the inverse relationship.

When to use it

Use inverse variation when one variable increases as the other decreases, and their product stays the same.

Level

Usually taught in: Algebra I

Worked examples

1. Find y using a simple inverse variation

Problem

Given

y = \frac{12}{x}

, find

y

when

x = 3

$y = \frac{12}{x}$
Start with the inverse variation equation $y = \frac{12}{x}$ where $k = 12$ .
$y = \frac{12}{3}$
Substitute $x = 3$ into the equation.
$y = 4$
Divide: $12 \div 3 = 4$ .

Answer: $y = 4$

When you know the value of the independent variable $x$ , substitute it directly into $y = \frac{k}{x}$ and simplify by dividing.

2. Solve for the independent variable using inverse variation

Problem

Given

y = \frac{20}{x}

, find

x

when

y = -5

$-5 = \frac{20}{x}$
Substitute $y = -5$ into the equation $y = \frac{20}{x}$ .
$-5x = 20$
Multiply both sides by $x$ to clear the fraction on the right side.
$x = \frac{20}{-5}$
Divide both sides by $-5$ to isolate $x$ .
$x = -4$
Compute: $20 \div (-5) = -4$ .

Answer: $x = -4$

When the variable is in the denominator, multiply both sides by that variable to move it to the numerator, then solve. Negative values can appear in inverse variation, so be careful with signs.

3. Apply inverse variation to a real-world scenario

Problem

A coffee shop discovers that the number of customers each day is inversely proportional to the outside temperature. When it is

40°\text{F}

, 80 customers visit. How many customers visit when the temperature is

50°\text{F}

$k = x \cdot y = 40 \cdot 80 = 3200$
In inverse variation, the product $xy$ is constant. Use the first pair of values to find $k$ .
$y = \frac{3200}{x}$
Write the inverse variation equation using the constant $k = 3200$ .
$y = \frac{3200}{50}$
Substitute $x = 50$ (the new temperature) into the equation.
$y = 64$
Divide: $3200 \div 50 = 64$ customers.

Answer: $y = 64$

Word problems require you to first identify the constant $k$ using a known pair of values, then use that constant to find unknown values. This method works for any pair of inversely related quantities.

Common mistakes

Where Inverse Variation usually goes wrong

The mistake	Why it is wrong	The fix
Writing $y = k \cdot x$ instead of $y = \frac{k}{x}$ .	Multiplying by $x$ creates direct variation, not inverse variation; the variables move in the same direction instead of opposite directions.	Remember: inverse means one variable is in the denominator. The formula must be $y = \frac{k}{x}$ so that as $x$ grows, $y$ shrinks.
Forgetting that $x$ cannot be zero when you try to use $x = 0$ in a problem.	Division by zero is undefined, so $x = 0$ has no meaning in inverse variation; the relationship breaks down at that value.	Always note that $x \neq 0$ (and $y \neq 0$ if $k \neq 0$ ). If a problem seems to require $x = 0$ , re-read it—the scenario is not actually inverse variation.
Using the wrong operation to find the constant $k$ , such as computing $x + y$ or $x - y$ instead of $x \cdot y$ .	The constant in inverse variation is always the product $xy$ , not the sum, difference, or any other combination; using a different operation gives the wrong formula.	Always verify: $k = x \cdot y$ for every pair of values. If one pair gives $k = 24$ but another gives $k = 15$ , the relationship is not inverse variation.

Tips and when to use something else

Write the inverse variation formula $y = \frac{k}{x}$ as soon as you identify the relationship, using the constant $k$ from the problem.
Check your constant by computing $xy$ for at least two pairs; if you get different products, the relationship is not inverse variation—use direct variation ( $y = kx$ ) instead.
Remember the graph of $y = \frac{k}{x}$ is a hyperbola, not a line; it gets very steep near $x = 0$ and flattens out far from the origin.
If a problem gives you one pair of values $(x_1, y_1)$ , always find the constant first: $k = x_1 \cdot y_1$ , then use $y = \frac{k}{x}$ for any other pair.

Related formulas

Direct VariationDirect variation describes how one quantity changes proportionally with another, letting you find unknown values when two variables stay linked by a constant ratio.RatioA ratio compares two quantities to show their relationship, and can be written as a fraction or with a colon to help you scale or find missing values.ProportionUse Proportion when two ratios are equal to find a missing value by cross-multiplying to solve equations with fractions.Function NotationFunction notation y = f(x) is a way to write and evaluate functions, showing how input values transform into output values using a concise symbolic format.Domain and RangeDomain lists all valid inputs for a function; range lists all possible outputs—use this to understand what a function can do.

Frequently asked questions

What is the difference between inverse and direct variation?: In direct variation, $y = kx$ , and both variables change in the same direction: if $x$ doubles, $y$ doubles. In inverse variation, $y = \frac{k}{x}$ , and the variables change in opposite directions: if $x$ doubles, $y$ is cut in half. Both have a constant, but direct variation multiplies and inverse variation divides.
How do I find the constant $k$ in an inverse variation?: Multiply the two known values together: $k = x \cdot y$ . For example, if $x = 5$ and $y = 8$ satisfy the relationship, then $k = 5 \cdot 8 = 40$ , and the equation is $y = \frac{40}{x}$ .
Can $x$ or $y$ be negative in inverse variation?: Yes. Negative values work just like positive values in the formula $y = \frac{k}{x}$ . If $k$ is positive and $x$ is negative, then $y$ will be negative. If both $x$ and $y$ are negative, then $k$ is positive. Just be careful with signs when doing arithmetic.
Why does the graph of $y = \frac{k}{x}$ never touch the axes?: The graph never crosses the $x$ -axis because $y$ can never equal zero (unless $k = 0$ , which is not inverse variation). Similarly, the graph never crosses the $y$ -axis because $x$ can never equal zero. The axes are called asymptotes—the curve approaches them but never reaches them.

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Reviewed 2026-09-18