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The magic of **mathematics** unfolds as we delve into the fascinating world of **direct variation equations**. Often encountered in academic and real-world situations, this equation **paints** a **vivid picture** of relationships where one variable changes directly from another.

It helps us explain **phenomena** like the **speed** of a car relating to the **time** it takes to reach a destination or how the **cost** of goods changes with **quantity**. The **mathematical simplicity** of **direct variation** belies its profound implications, giving us a glimpse into the **predictable** and **proportional relationships** that govern our world.

Dive into this article to unravel the** mystery**, **utility**, and **beauty** of **direct variation equations** and understand how they shape our understanding of the world.

## Definition of Direct Variation Equation

A **direct variation equation** is a mathematical relationship between two variables where the **ratio** of the two remains constant. In other words, when one variable increases, the other increases proportionally, and vice versa.

This relationship is often expressed in the form **y = kx**, where **y** and **x** are the two variables, and **k** is the **constant of variation**. If you know the value of one variable, you can use the direct variation equation to find the value of the other variable.

The constant **k** is determined by the specific relationship between the two variables. **Direct variation equations** play a vital role in understanding various **real-world scenarios**, from **physics** to **economics**, allowing for **predictable** and **consistent** calculations and predictions.

Below is the figure-1, we present a generic representation of the direct variation equations.

Figure-1.

**Historical Significance**

The concept of **direct variation**, also known as **direct proportion**, has roots in ancient mathematics. The principle of direct variation underpins the concept of **ratio** and **proportion**, which were pivotal in the mathematical systems of ancient civilizations.

**Ancient Babylonians (c. 1800 BCE)**

**Babylonian mathematics** used **ratios** extensively, though their understanding of **proportion** didn’t exactly equate to our modern concept of **direct variation**. They used ratios to solve problems related to trade, land measurement, and geometric calculations.

**Ancient Greeks (c. 300 BCE)**

In his work **“Elements,”** **Greek mathematician Euclid** formalized the concept of ratios and proportions. His definition of equal ratios can be seen as an early exploration of **direct variation**.

**Middle Ages and Renaissance**

During this period, **European mathematicians** translated **Greek** and **Arabic mathematical** texts and incorporated their knowledge, further developing concepts of **ratios** and **proportions**.

**17th Century**

With the advent of **analytical geometry** and calculus by mathematicians like **René Descartes** and **Isaac Newton**, the concept of **direct variation** became clearer. **Descartes** introduced the concept of representing equations **graphically**, which led to the understanding that direct variation equations represent **straight lines** passing through the **origin**.

**19th Century Onwards**

As mathematics became more abstract, the concept of **functions** was formalized. The direct variation was then seen as a specific type of function – a **linear function** with a **y-intercept** of **0**.

The concept of **direct variation** has been refined and abstracted over **millennia**, playing a fundamental role in our understanding of the **mathematical world**. It’s crucial in various fields such as **physics**, **economics**, **biology**, and **engineering**, demonstrating the power of this relatively simple **mathematical relationship**.

**Properties**

The properties of the direct variation equation are:

**Constant Ratio**

In a **direct variation**, the ratio of the two variables is always constant. This means that for any **two pairs** of corresponding values of **‘x’** and** ‘y’** **(x1, y1)** and (**x2, y2)**, the ratio **y1/x1** equals **y2/x2**.

**Origin Passage**

A **direct variation equation** always passes through the origin **(0,0)** in a graph. This is because when one variable is **zero**, the other is also **zero**. Therefore, the graph of a direct variation is a **straight line** passing through the origin.

**Proportional Change**

The variables in a direct variation change proportionally. If **‘x’ doubles**, **‘y’ doubles**; if **‘x’ triples**, **‘y’ triples**, and so on. Similarly, if **‘x’ is halved**, **‘y’ is also halved**. This is because of the constant ratio (**k**) between **‘x’** and** ‘y’**.

**Nonzero Constant**

The variation **‘k’ constant** is never **zero** in a direct variation. If **‘k’ were zero**, the equation would be **y = 0**, which does not represent a variable relationship.

**Unit Rate**

The constant of variation **‘k’** can also be considered as the **unit rate** or **slope** of the line in the **coordinate plane**. It represents the amount by which **‘y’** changes for every unit change in **‘x’**.

**Inverse Relationship With Reciprocal**

While in a **direct variation**, **y** varies directly as **x**, in its reciprocal form,** x** varies directly as **1/y**. This relationship solves many problems where the **direct variation** relationship is known, but the **inverse relationship** is required.

**Applications **

**Direct variation equations** are used in various disciplines to describe relationships where changes in one quantity lead to **proportional** changes in another. Here are some examples:

**Physics and Engineering**

**Direct variation** is fundamental to many **laws of physics** and principles of **engineering**. For instance, **Ohm’s Law** states that the **current** through a conductor between two points is directly proportional to the **voltage** across the two points. Here, the constant of proportionality is the **resistance**.

**Chemistry**

In gas laws, the **volume** of a gas varies directly with its **temperature** when **pressure** and the **amount of gas** are kept constant (**Charles’s Law**). Also, the **pressure** of a gas varies directly with its **temperature** when the **volume** and the **amount of gas** are kept constant (**Gay-Lussac’s Law**).

**Economics**

**Direct variation equations** are used in economics to describe the relationship between **supply** and **demand**, the effect of **interest rates** on investment, the **cost of goods** based on quantity, and much more. For example, the **total cost** of manufacturing ‘n’ items is usually a direct variation of** ‘n’**.

**Biology**

In **biology**, **direct variation** can explain phenomena like the relationship between the **metabolic rate** of an organism and its **body mass**.

**Computer Science**

In **computer science**, **direct variation** describes **algorithm complexity** in **Big O notation**. For example, in an **O(n)** algorithm, the time it takes to execute it varies directly with the input data size.

**Astronomy**

**Kepler’s Third Law** states that the square of the period of revolution of a **planet** around the sun is **directly proportional** to the cube of its average distance from the **sun**.

**Exercise **

**Example 1**

If **y** varies directly as** x**, and y = 20 when **x = 5**, find the** variation (k)** constant and write the variation equation.

**Solution**

From the equation **y = kx**, we can solve for **k** as:

y/x = k

When **y = 20** and **x = 5**, we find:

k = 20/5

k = 4

So, the **equation of variation** is **y = 4 * x**.

Figure-2.

**Example 2**

The **weight** of an **object** varies directly with its **volume**. If an object with a volume of **3** cubic meters weighs **6** kg, how much would an object with a volume of **7** cubic meters weigh?

**Solution**

The equation for this direct variation is:

Weight = k * Volume

When **volume = 3, **we get:

weight = 6

and

k = 6 / 3

k = 2

Therefore, the weight of an object with a volume of **7 cubic meters** would be **2 * 7 = 14 kg**.

**Example 3**

The amount of **money** (**M**) you earn **varies** directly with the number of hours (**h**) you work. If you earn **$150** for **10** hours of work, what is the **equation of variation**?

**Solution**

Here, **M = k * h**. Since **M = $150** when h = 10, we get:

k = 150 / 10

k = 15

So, the **equation of variation** is **M = 15 h**.

**Example 4**

Using the previous example, how much money would you earn for **15 work hours**?

**Solution**

With the equation **M = 15 h**, you would earn:

M = 15 * 15

M = $225 for 15 hours of work

**Example 5**

The **force of gravity** acting on an object is **directly proportional** to its **mass**. If an object with a mass of **2 kg** has a **gravitational force** of **19.6 N**, what would be the **gravitational force** acting on an object of mass 5 kg?

**Solution**

The equation for this direct variation is:

F = k * m

With F = 19.6 N when m = 2 kg, we find:

k = 19.6 / 2

k = 9.8

So, the **gravitational force** on an object of mass **5 kg** would be **9.8 * 5 = 49 N**.

**Example 6**

The cost of **apples (C)** varies directly with the number of apples **bought (n)**. If **4** apples cost **$8**, how much would** 10** apples cost?

**Solution**

In this direct variation:

C = k * n

When n = 4, C = $8, we get:

k = 8 / 4

k = 2

Therefore, the cost of **10 apples** would be **2 * 10 = $20**.

**Example 7**

The **distance (d)** a car travels varies directly with the** time (t)** it** travels**. If a car travels **300 km** in** 5 hours**, what is the **equation of variation**?

**Solution**

Here, **d = k * t**. Since **d = 300 km** when **t = 5** hours, we get

k = 300 / 5

k = 60

Therefore, the **equation of variation** is** d = 60 t**.

Figure-3.

**Example 8**

Using the previous example, how far would the car travel in **8 hours**?

**Solution**

Using the equation **d = 60 t**, the car would travel:

d = 60 * 8

d = 480 km in 8 hours

*All images were created with MATLAB.*