Reference

Variations

Concept

Constant of Variation

A constant of variation, also known as a constant of proportionality, is a non-zero constant that relates two variables.

Direct Variation

In a direct variation, the constant of variation

k

represents the ratio of two variables. The ratio remains unchanged when the variables

x

and

y

change.

\frac{y}{x} = k, where x \neq = 0 and y \neq = 0

When the variable

y

is isolated on the left-hand side, the constant of variation

k

defines the slope of the line through the origin that represents the relationship between

x

and

y .

\frac{y}{x} = k \Leftrightarrow y = k x

Inverse Variation

In an inverse variation, the constant of variation $k$ is the product of two variables. The product remains the same when the variables $x$ and $y$ change.

x y = k, where x \neq = 0 and y \neq = 0

Concept

Direct Variation

Direct variation, also known as direct proportionality or proportional relationship, occurs when two variables, $x$ and $y,$ have a relationship that forms a linear function passing through the origin where $x \neq = 0$ and $y \neq = 0 .$

$y = k x$

The constant

k

is the constant of variation. It defines the slope of the line. When

k = 0,

the relationship is not in direct variation. In the example below, the constant of variation is

k = 1.5 .

A line y=1.5x with a point on the line that can be moved

The constant of variation may be any real number except

0 .

It is worth noting that the quotient of

y

and

x

is the constant of variation.

y = k x \Leftrightarrow \frac{y}{x} = k

Here are some examples.

Examples of Direct Variation
Example	Rule	Comment
The circumference of a circle.	$C = π d$	Here, $d$ is the diameter of the circle and the constant of variation is $π .$
The mass of an object.	$m = ρ V$	Here, $ρ$ is the constant density of the object and $V$ is the volume.
Distance traveled at a constant rate.	$d = r t$	The constant of variation $r$ is the rate and $t$ is the time spent traveling.

Direct variation is closely related to other types of variation.

Concept

Inverse Variation

An inverse variation, or inverse proportionality, occurs when two non-zero variables have a relationship such that their product is constant. This relationship is often written with one of the variables isolated on the left-hand side.

$x y = k or y = \frac{k}{x}$

The constant

k

is the constant of variation. When

k = 0,

the relationship is not an inverse variation. In the following example, the constant of variation is

k = 2 .

A graph of a function y=2/x with a point on the graph that can be moved

The constant of variation may be any real number except

0 .

Here are some examples of inverse variations.

Examples of Inverse Variation
Example	Rule	Comment
The gas pressure in a sealed container if the container's volume is changed, given constant temperature and constant amount of gas.	$P = \frac{n R T}{V}$	The variables are the pressure $P$ and the volume $V .$ The amount of gas $n,$ temperature $T,$ and universal gas constant $R$ are fixed values. Therefore, the constant of variation is $n R T .$
The time it takes to travel a given distance at various speeds.	$t = \frac{d}{s}$	The constant of variation is the distance $d$ and the variables are the time $t$ and the speed $s .$

Inverse variation is closely related to other types of variation.

Rule

Product Rule for Inverse Variations

If the ordered pairs $(x_{1}, y_{1})$ and $(x_{2}, y_{2})$ are solutions to an inverse variation, then the products $x_{1} y_{1}$ and $x_{2} y_{2}$ are equal.

y_{1} = \frac{k}{x _{1}} and y_{2} = \frac{k}{x _{2}} ⇓ x_{1} y_{1} = x_{2} y_{2}

Proof

(x_{1}, y_{1})

and

(x_{2}, y_{2})

are solutions to an inverse variation with constant of variation

k,

then both of the products

x_{1} y_{1}

and

x_{2} y_{2}

are equal to

k .

y_{1} = \frac{k}{x _{1}} y_{2} = \frac{k}{x _{2}} \Rightarrow x_{1} y_{1} = k \Rightarrow x_{2} y_{2} = k

Since the products

x_{1} y_{1}

and

x_{2} y_{2}

are both equal to the same constant, by the Transitive Property of Equality they are also equal to each other.

x_{1} y_{1} = x_{2} y_{2}

Concept

Joint Variation

A joint variation, also known as joint proportionality, occurs when one variable varies directly with two or more variables. In other words, if a variable varies directly with the product of other variables, it is called joint variation.

$z = k x y$

Here, the variable $z$ varies jointly with $x$ and $y,$ and $k$ is the constant of variation. Here are some examples of joint variation.

Examples of Joint Variation
Example	Rule	Comment
The area of a rectangle	$A = ℓ w$	Here, $ℓ$ is the rectangle's length, $w$ its width, and the constant of variation $k$ is $1 .$
The volume of a pyramid	$V = \frac{1}{3} ℓ w h$	Here, $ℓ$ and $w$ are the length and the width of the base, respectively, while $h$ is the pyramid's height. The constant of variation $k$ is $\frac{1}{3} .$

Lastly, it is important to note that joint variation is closely related to other types of variation.

Concept

Combined Variation

A combined variation, or combined proportionality, occurs when one variable depends on two or more variables, either directly, inversely, or a combination of both. This means that any joint variation is also a combined variation.

$z = \frac{k x}{y}$

The variable $z$ varies directly with $x$ and inversely with $y,$ and $k$ is the constant of variation. Therefore, this is a combined variation. Here are some examples.

Examples of Combined Variation
Example	Rule	Comment
Newton's Law of Gravitational Force	$F = \frac{G m _{1} m _{2}}{d ^{2}}$	The gravitational force $F$ varies directly as the masses of the objects $m_{1}$ and $m_{2},$ and inversely as the square of the distance $d^{2}$ between the objects. The gravitational constant $G$ is the constant of variation.
The Ideal Gas Law	$P = \frac{n R T}{V}$	The pressure $P$ varies directly as the number of moles $n$ and the temperature $T,$ and inversely as the volume $V .$ The universal gas constant $R$ is the constant of variation.

Combined variation is closely related to other types of variation.

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Direct Variation

Inverse Variation

Proof