\[$ Y \$\] Varies Directly With \[$ X \$\], And \[$ Y = 21 \$\] When \[$ X = 15 \$\].1. Variation Equation: \[$ Y = Kx \$\], Where \[$ K \$\] Is The Constant Of Variation. - Given \[$ Y = 21

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Direct Variation: Understanding the Relationship Between Two Variables

Introduction

In mathematics, direct variation is a fundamental concept that describes the relationship between two variables. It is a type of linear relationship where one variable is a constant multiple of the other variable. In this article, we will explore the concept of direct variation, its equation, and how to find the constant of variation. We will also use a real-world example to illustrate the concept.

What is Direct Variation?

Direct variation is a relationship between two variables, x and y, where y is a constant multiple of x. This means that as x increases or decreases, y also increases or decreases at a constant rate. The relationship can be represented by the equation y = kx, where k is the constant of variation.

The Equation of Direct Variation

The equation of direct variation is y = kx, where k is the constant of variation. This equation states that y is equal to k times x. The value of k determines the rate at which y changes in response to changes in x.

Finding the Constant of Variation

To find the constant of variation, we need to know the values of x and y at a specific point. Let's say we know that y = 21 when x = 15. We can use this information to find the value of k.

Step 1: Write the Equation of Direct Variation

The equation of direct variation is y = kx. We know that y = 21 when x = 15, so we can substitute these values into the equation:

21 = k(15)

Step 2: Solve for k

To solve for k, we need to isolate k on one side of the equation. We can do this by dividing both sides of the equation by 15:

k = 21/15

k = 1.4

Step 3: Write the Equation of Direct Variation with the Value of k

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

y = 1.4x

Real-World Example

Let's say we are a manager of a company that produces widgets. We know that the number of widgets produced per hour is directly proportional to the number of machines operating. If we have 15 machines operating and produce 21 widgets per hour, we can use the equation of direct variation to find the number of widgets produced per hour if we have 20 machines operating.

Step 1: Write the Equation of Direct Variation

The equation of direct variation is y = kx. We know that y = 21 when x = 15, so we can substitute these values into the equation:

21 = k(15)

Step 2: Solve for k

To solve for k, we need to isolate k on one side of the equation. We can do this by dividing both sides of the equation by 15:

k = 21/15

k = 1.4

Step 3: Write the Equation of Direct Variation with the Value of k

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

y = 1.4x

Step 4: Find the Number of Widgets Produced per Hour with 20 Machines Operating

Now that we have the equation of direct variation, we can use it to find the number of widgets produced per hour if we have 20 machines operating. We can substitute x = 20 into the equation:

y = 1.4(20)

y = 28

Therefore, if we have 20 machines operating, we can produce 28 widgets per hour.

Conclusion

In this article, we have explored the concept of direct variation, its equation, and how to find the constant of variation. We have also used a real-world example to illustrate the concept. Direct variation is a fundamental concept in mathematics that describes the relationship between two variables. It is a type of linear relationship where one variable is a constant multiple of the other variable. By understanding direct variation, we can solve problems in a variety of fields, including business, economics, and science.

Key Takeaways

  • Direct variation is a relationship between two variables, x and y, where y is a constant multiple of x.
  • The equation of direct variation is y = kx, where k is the constant of variation.
  • To find the constant of variation, we need to know the values of x and y at a specific point.
  • We can use the equation of direct variation to solve problems in a variety of fields, including business, economics, and science.

Frequently Asked Questions

  • What is direct variation? Direct variation is a relationship between two variables, x and y, where y is a constant multiple of x.
  • What is the equation of direct variation? The equation of direct variation is y = kx, where k is the constant of variation.
  • How do I find the constant of variation? To find the constant of variation, we need to know the values of x and y at a specific point.
  • What are some real-world applications of direct variation? Direct variation has a variety of real-world applications, including business, economics, and science.
    Direct Variation: Frequently Asked Questions

Introduction

In our previous article, we explored the concept of direct variation, its equation, and how to find the constant of variation. In this article, we will answer some frequently asked questions about direct variation.

Q: What is direct variation?

A: Direct variation is a relationship between two variables, x and y, where y is a constant multiple of x. This means that as x increases or decreases, y also increases or decreases at a constant rate.

Q: What is the equation of direct variation?

A: The equation of direct variation is y = kx, where k is the constant of variation. This equation states that y is equal to k times x.

Q: How do I find the constant of variation?

A: To find the constant of variation, we need to know the values of x and y at a specific point. We can use the equation y = kx to solve for k.

Q: What are some real-world applications of direct variation?

A: Direct variation has a variety of real-world applications, including business, economics, and science. For example, the number of widgets produced per hour is directly proportional to the number of machines operating.

Q: Can I use direct variation to solve problems in other fields?

A: Yes, direct variation can be used to solve problems in a variety of fields, including business, economics, and science. It is a fundamental concept in mathematics that describes the relationship between two variables.

Q: How do I know if a problem involves direct variation?

A: To determine if a problem involves direct variation, look for a relationship between two variables where one variable is a constant multiple of the other variable. If you see this type of relationship, you can use the equation of direct variation to solve the problem.

Q: Can I use direct variation to solve problems with more than two variables?

A: While direct variation is typically used to solve problems with two variables, it can also be used to solve problems with more than two variables. However, the relationship between the variables must still be linear.

Q: What are some common mistakes to avoid when using direct variation?

A: Some common mistakes to avoid when using direct variation include:

  • Assuming that the relationship between the variables is linear when it is not.
  • Failing to check for extraneous solutions.
  • Not using the correct equation of direct variation.

Q: How do I check for extraneous solutions?

A: To check for extraneous solutions, plug the solution back into the original equation and check if it is true. If it is not true, then the solution is extraneous.

Q: Can I use direct variation to solve problems with negative values?

A: Yes, direct variation can be used to solve problems with negative values. However, you must be careful when working with negative values to ensure that you are using the correct equation of direct variation.

Conclusion

In this article, we have answered some frequently asked questions about direct variation. Direct variation is a fundamental concept in mathematics that describes the relationship between two variables. It is a type of linear relationship where one variable is a constant multiple of the other variable. By understanding direct variation, you can solve problems in a variety of fields, including business, economics, and science.

Key Takeaways

  • Direct variation is a relationship between two variables, x and y, where y is a constant multiple of x.
  • The equation of direct variation is y = kx, where k is the constant of variation.
  • To find the constant of variation, we need to know the values of x and y at a specific point.
  • Direct variation has a variety of real-world applications, including business, economics, and science.
  • To check for extraneous solutions, plug the solution back into the original equation and check if it is true.

Frequently Asked Questions

  • What is direct variation? Direct variation is a relationship between two variables, x and y, where y is a constant multiple of x.
  • What is the equation of direct variation? The equation of direct variation is y = kx, where k is the constant of variation.
  • How do I find the constant of variation? To find the constant of variation, we need to know the values of x and y at a specific point.
  • What are some real-world applications of direct variation? Direct variation has a variety of real-world applications, including business, economics, and science.