Direct, Inverse and Joint Variation

Quick answer

Two quantities vary directly when y = kx: doubling one doubles the other, the ratio y/x stays equal to k, and the graph is a line through the origin. They vary inversely when y = k/x: doubling one halves the other, and the product xy stays equal to k. Joint variation involves a product of several variables, as in z = kxy, and combined variation mixes the two, as in y = kx/z². In every case one known set of values gives k, and k answers everything else.

What you'll learn

Write and use direct variation equations
Write and use inverse variation equations
Model joint and combined variation
Decide from a table whether a relationship is direct or inverse

Direct variation

A car at a steady $60$ miles per hour covers $60$ , $120$ and $180$ miles in $1$ , $2$ and $3$ hours. Distance varies directly with time:

y = kx \qquad k \ne 0

The constant of variation $k$ is the ratio $\tfrac{y}{x}$ , the same for every pair. The graph is a line through the origin with slope $k$ .

Inverse variation

A trip of $180$ miles takes $3$ hours at $60$ mph, $2$ hours at $90$ mph and $4$ hours at $45$ mph. Time varies inversely with speed:

y = \frac{k}{x}

Here the product $xy$ is the constant: speed times time is always $180$ .

y = 12/x

Inverse variation: y = 12/x

Why one ratio or one product decides everything

In direct variation, $\tfrac{y}{x} = k$ is fixed, so multiplying $x$ by any factor multiplies $y$ by the same factor. In inverse variation, $xy = k$ is fixed, so multiplying $x$ by a factor divides $y$ by it. Every point on the graph of $y = \tfrac{12}{x}$ is the corner of a rectangle of area $12$ . Direct variation keeps a ratio constant and inverse variation keeps a product constant, so one pair of values finds $k$ , and $k$ finds every other value.

Joint and combined variation

A quantity can vary with several others at once.

Joint variation: $z = kxy$ . The area of a triangle varies jointly with its base and height, with $k = \tfrac{1}{2}$ .
Combined variation: direct and inverse together, as in $y = \tfrac{kx}{z^2}$ . Newton’s law of gravity, $F = \tfrac{Gm_1m_2}{d^2}$ , varies jointly with the masses and inversely with the square of the distance.

Worked examples

Example 5 — Reading a table

Decide whether each table shows direct or inverse variation.

$x$	$2$	$4$	$8$
$y$	$12$	$6$	$3$

The products $xy$ are all $24$ , so $y$ varies inversely with $x$ : $y = \tfrac{24}{x}$ . Had the ratios $\tfrac{y}{x}$ been equal instead, the variation would be direct.

Common mistakes

Practice problems

$y$ varies directly with $x$ , and $y = 15$ when $x = 5$ . Find $y$ when $x = 8$ .

Answer

$24$

Full solution

$k = 3$ , so $y = 3 \cdot 8$ .
$y$ varies inversely with $x$ , and $y = 8$ when $x = 3$ . Find $y$ when $x = 6$ .

Answer

$4$

Full solution

$k = 24$ , so $y = \tfrac{24}{6}$ .
Is $y = 4x + 1$ a direct variation?

Answer

No

Full solution

Its graph does not pass through the origin: at $x = 0$ , $y = 1$ . The ratio $\tfrac{y}{x}$ changes from point to point.
In a table, $x = 2, 4, 8$ go with $y = 12, 6, 3$ . Write the equation.

Answer

$y = \tfrac{24}{x}$

Full solution

Every product $xy$ is $24$ : inverse variation.
$z$ varies jointly with $x$ and $y$ , and $z = 24$ when $x = 2$ and $y = 3$ . Find $z$ when $x = 5$ and $y = 1$ .

Answer

$20$

Full solution

$24 = 6k$ , so $k = 4$ , and $z = 4 \cdot 5 \cdot 1$ .
The area of a triangle varies jointly with its base and height. What is the constant of variation?

Answer

$\tfrac{1}{2}$

Full solution

$A = \tfrac{1}{2}bh$ has the form $A = kbh$ with $k = \tfrac{1}{2}$ .
Gas pressure varies inversely with volume. A gas at $300$ kPa fills $2$ liters. What is the pressure at $5$ liters?

Answer

$120$ kPa

Full solution

$k = 300 \cdot 2 = 600$ , so $P = \tfrac{600}{5}$ .
$y$ varies directly with the square of $x$ , and $y = 18$ when $x = 3$ . Find $y$ when $x = 5$ .

Answer

$50$

Full solution

$y = kx^2$ with $18 = 9k$ , so $k = 2$ and $y = 2 \cdot 25$ .
Light intensity varies inversely with the square of the distance. It is $90$ lux at $2$ meters. What is it at $6$ meters?

Answer

$10$ lux

Full solution

$k = 90 \cdot 2^2 = 360$ , so $I = \tfrac{360}{36}$ . Tripling the distance divides the intensity by $9$ .
$y$ varies inversely with $x$ , and $y = 10$ when $x = 2$ . A student reasons that $y$ falls as $x$ grows, writes the line $y = -5x + 20$ , and predicts $y = 0$ at $x = 4$ . What went wrong?

Hint

What stays constant in inverse variation?

Answer

Inverse variation keeps $xy$ constant: $y = \tfrac{20}{x}$ , so $y = 5$ at $x = 4$ .

Full solution

A decreasing line is not inverse variation. With $k = 2 \cdot 10 = 20$ , doubling $x$ from $2$ to $4$ halves $y$ from $10$ to $5$ . The value of $y$ never reaches $0$ .

Frequently asked questions

What is direct variation?

A relationship y = kx with a nonzero constant k. The ratio y/x is always k, and the graph is a line through the origin.

What is inverse variation?

A relationship y = k/x. The product xy is always k, so when one quantity doubles the other halves.

What is joint variation?

When one quantity varies directly with the product of two or more others, as in z = kxy.

How do I find the constant of variation?

Substitute one known set of values into the equation and solve for k. For y = kx with y = 12 when x = 4, k = 3.

Is every line direct variation?

No. Only lines through the origin. y = 2x + 3 is linear, but y/x is not constant, so it is not direct variation.

What to learn next

Graphing Rational Functions: Asymptotes, Holes and ZerosHow factoring a rational function reveals its zeros, holes and vertical asymptotes, why the graph shoots off near an asymptote, and how degrees decide the end behavior.

Standards alignment

This lesson covers the following Common Core State Standards for Mathematics.

CCSS.MATH.CONTENT.HSA.CED.A.2Creating EquationsCreate equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.

For teachersPrint problems onlyTick to print a blank worksheet; leave it clear to print the lesson with the full answer key.

Quick answer

What you'll learn

Direct variation

Inverse variation

Why one ratio or one product decides everything

Joint and combined variation

Worked examples

Common mistakes

Practice problems

Frequently asked questions

What is direct variation?

What is inverse variation?

What is joint variation?

How do I find the constant of variation?

Is every line direct variation?

What to learn next

Standards alignment

More Algebra 2 lessons