Write the Equation of Variation

Oct 13, 2025 by miodeya.com 25 views

Introduction

In mathematics, the concept of variation is a fundamental idea that helps us understand the relationships between different quantities. When two quantities vary directly or inversely with each other, we can use the equation of variation to describe their relationship. In this article, we will explore the concept of variation and derive the equation of variation for a given problem.

What is Variation?

Variation is a mathematical concept that describes the relationship between two or more quantities. When two quantities vary directly with each other, it means that as one quantity increases, the other quantity also increases at a constant rate. On the other hand, when two quantities vary inversely with each other, it means that as one quantity increases, the other quantity decreases at a constant rate.

Direct Variation

Direct variation is a type of variation where two quantities vary directly with each other. The equation of direct variation is given by:

p = kq

where p is the dependent variable, q is the independent variable, and k is the constant of variation.

Inverse Variation

Inverse variation is a type of variation where two quantities vary inversely with each other. The equation of inverse variation is given by:

p = k/r

where p is the dependent variable, r is the independent variable, and k is the constant of variation.

The Problem

The problem states that the quantity p varies directly with q^2 and inversely with r. When q = -3 and r = 3, p = -33. We need to find the equation of variation.

Step 1: Write the Equation of Variation

Since p varies directly with q^2 and inversely with r, we can write the equation of variation as:

p = k(q^2)/r

Step 2: Substitute the Given Values

We are given that q = -3 and r = 3. We can substitute these values into the equation of variation:

-33 = k((-3)^2)/3

Step 3: Simplify the Equation

We can simplify the equation by evaluating the exponent:

-33 = k(9)/3

Step 4: Solve for k

We can solve for k by multiplying both sides of the equation by 3:

-99 = 9k

Step 5: Divide Both Sides by 9

We can divide both sides of the equation by 9 to solve for k:

k = -99/9

k = -11

Step 6: Write the Final Equation of Variation

Now that we have found the value of k, we can write the final equation of variation:

p = (-11(q^2))/r

Conclusion

In this article, we explored the concept of variation and derived the equation of variation for a given problem. We started by writing the equation of variation, substituted the given values, simplified the equation, solved for k, and finally wrote the final equation of variation. The equation of variation is a powerful tool that helps us understand the relationships between different quantities.

Final Answer

The final answer is:

Introduction

In our previous article, we explored the concept of variation and derived the equation of variation for a given problem. In this article, we will answer some frequently asked questions about variation to help you better understand this mathematical concept.

Q: What is variation?

A: Variation is a mathematical concept that describes the relationship between two or more quantities. When two quantities vary directly or inversely with each other, we can use the equation of variation to describe their relationship.

Q: What are the different types of variation?

A: There are two main types of variation: direct variation and inverse variation.

Direct Variation: When two quantities vary directly with each other, it means that as one quantity increases, the other quantity also increases at a constant rate. The equation of direct variation is given by: p = kq
Inverse Variation: When two quantities vary inversely with each other, it means that as one quantity increases, the other quantity decreases at a constant rate. The equation of inverse variation is given by: p = k/r

Q: How do I determine if two quantities vary directly or inversely?

A: To determine if two quantities vary directly or inversely, you need to examine the relationship between the two quantities. If the quantities increase or decrease at a constant rate, they vary directly. If the quantities increase or decrease at a constant rate, but in opposite directions, they vary inversely.

Q: What is the equation of variation?

A: The equation of variation is a mathematical equation that describes the relationship between two or more quantities. The equation of variation is given by:

Direct Variation: p = kq
Inverse Variation: p = k/r

Q: How do I find the equation of variation?

A: To find the equation of variation, you need to follow these steps:

Write the Equation of Variation: Write the equation of variation based on the type of variation (direct or inverse).
Substitute the Given Values: Substitute the given values into the equation of variation.
Simplify the Equation: Simplify the equation by evaluating any exponents or expressions.
Solve for k: Solve for k by isolating it on one side of the equation.
Write the Final Equation of Variation: Write the final equation of variation by substituting the value of k.

Q: What is the constant of variation?

A: The constant of variation is a value that is used in the equation of variation to describe the relationship between the two quantities. The constant of variation is denoted by the letter k.

Q: How do I use the equation of variation in real-life situations?

A: The equation of variation can be used in a variety of real-life situations, such as:

Physics: The equation of variation is used to describe the relationship between distance, velocity, and time.
Economics: The equation of variation is used to describe the relationship between supply and demand.
Engineering: The equation of variation is used to describe the relationship between different physical quantities, such as force and distance.

Conclusion

In this article, we answered some frequently asked questions about variation to help you better understand this mathematical concept. We covered topics such as the definition of variation, the different types of variation, and how to use the equation of variation in real-life situations. By understanding variation, you can better analyze and describe the relationships between different quantities.