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

$xy =k,wherex =0andy =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 $y.$
$xy =k⇔y=kx $

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.

$xy=k,wherex =0andy =0 $

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 $x =0$ and $y =0.$

$y=kx$

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=kx⇔xy =k $

Here are some examples. Examples of Direct Variation | ||
---|---|---|

Example | Rule | Comment |

The circumference of a circle. | $C=πd$ | Here, $d$ is the diameter of the circle and the constant of variation is $π.$ |

The mass of an object. | $m=ρV$ | Here, $ρ$ is the constant density of the object and $V$ is the volume. |

Distance traveled at a constant rate. | $d=rt$ | 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.

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.

$xy=kory=xk $

The constant of variation may be any real number except $0.$ Here are some examples of inverse variations.

Examples of Inverse Variation
| ||
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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=VnRT $ | 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 $nRT.$ |

The time it takes to travel a given distance at various speeds. | $t=sd $ | The constant of variation is the distance $d$ and the variables are the time $t$ and the speed $s.$ |

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}=x_{1}k andy_{2}=x_{2}k ⇓x_{1}y_{1}=x_{2}y_{2} $

If $(x_{1},y_{1})$ 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}=x_{1}k y_{2}=x_{2}k ⇒x_{1}y_{1}=k⇒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} $

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.

$z=kxy$

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 | $A=ℓw$ | Here, $ℓ$ is the rectangle's length, $w$ its width, and the constant of variation $k$ is $1.$ |

The volume of a pyramid | $V=31 ℓwh$ | 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 $31 .$ |

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=ykx $

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=d_{2}Gm_{1}m_{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=VnRT $ | 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. |