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**Inverse variation equations** represent a captivating class of mathematical relationships that embody a distinct form of **interdependence**. When an **inverse variation equation** links two variables, an increase in one leads to a corresponding **decrease** in the other, and vice versa.

This article will explore the **mathematical elegance** and **practical importance** inherent to inverse variation equations.

## Defining Inverse Variation Equation

An **inverse variation equation** is a relationship between two variables such that their product is a constant. It is also known as **inverse proportionality** or **inverse variation**. In mathematical terms, if two variables **x** and **y** are **inversely proportional**, it can be written as:

xy = k

where:

**x**and**y**are the**variables**,**k**is a non-zero**constant**.

This equation **expresses** that when** x increases**, **y decreases** so that their product remains the same, and vice versa. For instance, if you were to **double x**, you would need to **halve y** to maintain the same product **k**.

The **equation** can be rearranged to make y the subject:

y = k / x

This shows that** y** equals a** constant** divided by **x**, which underlines the idea of **inverse variation**: **y decreases** as **x increases**, and vice versa.

Figure-1.

**Inverse variation equations** describe many real-world phenomena where one quantity** decreases** when another **increases**, such as the speed required to travel a certain distance in a fixed amount of time or the light intensity as you move away from a **source**.

**Properties of ****Inverse Variation Equation**

An **inverse variation equation**, also known as an **inverse proportionality equation**, has several key properties. Understanding these characteristics can help you identify, work with, and apply inverse variation in various mathematical and real-world contexts.

**Constant Product**

The most fundamental property of **inverse variation** is that the product of the two variables is constant. If **y** varies inversely as **x**, then the product **xy** = **k**, where **k** is a non-zero **constant**. This means that as one **variable increases**, the other variable decreases so that its **product** remains constant.

**Inverse Relationship**

If **y**** varies inversely** as **x**, then as **x** increases, **y** decreases **proportionally**, and vice versa. This is where the term **“inverse variation”** comes from.

This relationship is represented by the equation **y** = **k** / **x**, where **k** is a non-zero constant. The equation** signifies** that as **x** increases, **y** decreases in a way that maintains the** inverse variation relationship**.

**Non-linearity**

Inverse variation is indeed a type of **non-linear relationship** between two variables. Unlike in a **linear relationship**, where the graph forms a straight line, the graph of an **inverse variation equation** forms a **hyperbola**.

A **hyperbola** is a curved shape consisting of two separate branches, a characteristic feature of **inverse variation**. The **hyperbolic nature** of the graph demonstrates the inverse relationship between the variables and highlights the **non-linear** behavior of the **equation**.

**Asymptotic behavior**

The graph of an **inverse variation equation** indeed has two **asymptotes**. The **x-axis** (where **y = 0**) and the **y-axis** (where **x = 0**) are asymptotes. This means that the graph approaches these axes but never crosses them.

As **x** or **y** approaches zero, the value of the other variable becomes **extremely large**, and the graph gets closer and closer to the respective **axis**. The presence of asymptotes adds to understanding the behavior and limits of the **inverse variation relationship**.

**Quadrants**

The **inverse variation equation graph** is located in the **first and third quadrants** when **k** is positive and in the **second and fourth quadrants** when **k** is negative. This is because the product of **x** and **y** is always **positive** in the first and third **quadrants** and **negative** in the second and fourth **quadrants**.

The sign of **k** determines the** orientation** of the graph and the regions where the inverse variation relationship holds. Understanding the **quadrants** where the graph lies provides insights into the nature and **characteristics** of the **inverse variation equation**.

**Zeroes**

An **inverse variation equation** has no zeroes, as no** x** or **y** values can make the **equation equal** to** zero**. That’s because the constant k in the equation **xy = k** is non-zero.

**Sign of Variables**

In an **inverse variation**, the sign of** x** and **y** will always be the same if **k** is **positive** and different if** k** is **negative**.

These properties of **inverse variation equations** apply not only to **abstract mathematical problems** but also to a wide range of **real-world scenarios**, from the physics of light and sound to various principles in **engineering** and **economics**.

**Ralevent Formulas**

In **inverse variation relationships**, there are a couple of key formulas that are generally used:

**Inverse Variation Formula**

The main formula that defines **inverse variation** between two variables, **x,** and **y**, is expressed as:

xy = k

Here,** k** is a **non-zero constant**.

**Reformatted Inverse Variation Formula**

You can also express the **inverse variation** formula by **isolating** one variable, typically written as:

y = k / x

Here,** y** varies inversely with** x**. When **x** increases, **y** decreases, and vice versa.

**Solving for the Constant, k**

In many problems, you’ll be given **x** and **y** and asked to find the **constant of variation**, **k**. You can solve for **k** by **rearranging** the formula:

k = xy

Once you’ve found **k**, you can use it to find **y** for any given **x** or **x** for any given **y**.

**Solving for Unknown Variables**

If you’re given **k** and either **x** or **y**, you can solve for the **unknown variable** by rearranging the formula:

If given** k** and **y**, solve for **x**:

x = k / y

If given **k** and **x**, solve for **y**:

y = k / x

**Exercise **

**Example 1**

If **y** varies **inversely** as **x**, and** y = 2** when** x = 3**, find the constant of variation, **k**.

### Solution

We can find k using the formula:

k = x * y

So:

k = 2 * 3

k = 6

**Example 2**

Given the **inverse variation equation** **x * y = 4**, find **y** when **x = 2**.

Figure-2.

### Solution

Solve the equation for **y** by dividing both sides by **x**:

y = 4 / x

Substitute** x = 2** we get:

y = 4 / 2

y = 2

**Example 3**

If **y** varies inversely as **x**, and **y = 5** when **x = 1**, what is **y** when **x = 10**?

### Solution

First, find the constant of variation:

k = xy

k = 5 * 1

k = 5

Then, using** y = k / x**, substitute **k = 5** and **x = 10**, we get:

y = 5 / 10

y = 0.5

**Example 4**

If** y** varies inversely as **x**, and **y = 3** when **x = 4**, find **x** when **y = 1**.

### Solution

First, find the constant of variation:

k = xy

k = 3 * 4

k = 12

Then, using **x = k / y**, substitute **k = 12** and **y = 1**, we get:

x = 12 / 1

x = 12

**Example 5**

If **y** varies inversely as** x**, and **y = 2** when **x = 5**, find **y** when **x = 15**.

### Solution

First, find the constant of variation:

k = xy

k = 2 * 5

k = 10

Then, using **y = k / x**, substitute **k = 10** and **x = 15**, we get:

y = 10 / 15

y = 2/3

y ≈ 0.67

**Example 6**

Given the **inverse variation equation** **x * y = 6**, find **x** when **y = 3**.

Figure-3.

### Solution

Solve the equation for **x** by dividing both sides by **y**:

x = 6 / y

Substitute y = 3, we get:

x = 6 / 3

x = 2

**Example 7**

If **y** varies inversely as **x**, and **y = 4** when **x = 3**, what is **y** when** x = 9**?

### Solution

First, find the constant of variation:

k = xy

k = 4 * 3

k = 12

Then, using **y = k / x**, substitute** k = 12** and **x = 9**, we get:

y = 12 / 9

y = 4/3

y ≈ 1.33

**Example 8**

If **y** varies inversely as **x**, and **y = 6** when** x = 2**, find **x** when** y = 3**.

### Solution

First, find the constant of variation:

k = xy

k = 6 * 2

k = 12

Then, using **x = k / y**, substitute** k = 12** and** y = 3**, we get:

x = 12 / 3

x = 4

**Applications**

**Physics – Gravitational and Electrostatic Forces**

Both **gravitational** and **electrostatic forces** follow an** inverse-square law**, meaning the force decreases with the square of the distance. If you double the distance between two masses or charges, the **gravitational** or **electrostatic force** becomes a quarter of what it was.

The equations are:

F₉₉₍G₎ = G(m₁m₂)/r²

and

Fₑₗₑ = k(q₁q₂)/r²

where **G** and **k** are constants, **m1,** and **m2** are the two masses, **q1** and **q2** are the two charges, and **r** is the distance between the **masses** or **charges**.

**Chemistry – Ideal Gas Law**

In the **ideal gas law**, the **pressure** **P** of a gas is inversely proportional to its **volume** **V**, assuming the **temperature** and **amount of gas** remain constant. This relationship is known as **Boyle’s Law**, and it is expressed as **PV = k**, where **k** is a constant.

According to **Boyle’s Law**, if a gas is **compressed** (decreasing volume), the **pressure increases**. Conversely, if a gas is allowed to **expand** (increasing volume), the **pressure decreases**.

**Economics – Demand and Supply**

The **law of demand** states that the **price** of a product is **inversely proportional** to the **quantity demanded**. In other words, as the price of a product **increases**, the **demand decreases**, and vice versa.

This law **reflects** the general consumer behavior where a higher price tends to **discourage consumers** from purchasing a larger quantity of the product, while a** lower price stimulates** **demand**.

**Biology – Metabolic Rate**

The relationship between an organism’s **body size** and **metabolic rate** often follows an **inverse variation**. For example, smaller animals tend to have **higher metabolic rates** than larger animals. Therefore, a **mouse** has a higher metabolic rate than an **elephant**.

**Astronomy – Kepler’s Third Law**

In **astronomy**, the **square of the period** of a planet’s orbit is **inversely proportional** to the **cube of the semi-major axis** of its orbit. This relationship is known as **Kepler’s Third Law** or the **law of periods**. It states that planets farther from the sun have **longer orbital periods**.

**Engineering – Signal Processing**

In **signal processing**, the** frequency** of a signal is **inversely proportional** to its **period**. If you increase the frequency of a signal, the period (or time taken for one complete cycle of the signal) decreases.

**Ecology – Animal Population Dynamics**

In **ecology**, the **carrying capacity**, representing the maximum population size an environment can sustain, often exhibits **inverse variation** with the **per capita growth rate**. This means that as an **animal population** approaches or exceeds the carrying capacity of its environment, the **per capita growth** rate decreases.

**Computer Science – Parallel Processing**

In **parallel computing**, the time taken to process a task can be **inversely proportional** to the number of processors, assuming perfectly parallelizable tasks. This relationship is based on the concept of **parallel speedup**. If you double the number of processors, the time taken to process the task has the potential to halve, thereby improving **computational efficiency.**

## Historical Significance

**Inverse variation**, also known as** inverse proportion** or** variation**, has a rich historical background, much of which is intertwined with the development of **mathematics** and **scientific thought** over centuries.

**Ancient Mathematics**

The concept of **inverse proportionality** can be traced back to **ancient mathematics**. **Ancient Greeks**, including mathematicians like **Euclid** and **Pythagoras**, studied **proportional relationships**, though it’s unclear if they specifically dealt with **inverse proportions**.

**Middle Ages and Renaissance**

The principle of **inverse variation** gained explicit recognition during the **Middle Ages** and **Renaissance** periods. Scholars in the **Middle East** and **Europe** began developing the foundations of **algebra**, which led to a better understanding of **proportional relationships**, including **inverse variation**.

**17th Century**

The formulation of the **inverse-square law** in **physics** in the **17th century** by **Sir Isaac Newton** was a crucial historical point for **inverse variation**. Newton’s **Law of Universal Gravitation** states that the **force of gravity** between two objects decreases with the **square** of the **distance** between them, a clear example of an **inverse variation** relationship.

**18th Century**

**Swiss mathematician Leonhard Euler** and others developed the **formal mathematical notation** and **language** we use today to describe **inverse variation** and many other mathematical relationships.

**19th and 20th Centuries**

In the **19th** and **20th centuries**, **inverse variation** became an important tool in numerous emerging scientific fields. For instance, it was used to describe phenomena such as **electrical resistance** in circuits, the behavior of **gases** in chemistry, and principles of **economics** like **supply** and **demand**.

In **mathematics**, **inverse variation** has emerged and evolved as part of the broader development of **mathematical thought**. From early explorations of **proportion** to the sophisticated mathematical languages of modern science, **inverse variation** has proven to be a vital tool in describing the world around us.

*All images were created with GeoGebra.*