What Is The Constant Of Variation, K K K , Of The Direct Variation Y = K X Y = Kx Y = K X Through The Point ( − 3 , 2 (-3, 2 ( − 3 , 2 ]?A. K = 3 2 K = \frac{3}{2} K = 2 3 ​ B. K = − 2 3 K = -\frac{2}{3} K = − 3 2 ​ C. K = 2 3 K = \frac{2}{3} K = 3 2 ​ D. K = 3 2 K = \frac{3}{2} K = 2 3 ​

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Introduction

In mathematics, direct variation is a type of relationship between two variables where one variable is a constant multiple of the other. This relationship is often represented by the equation y = kx, where y is the dependent variable, x is the independent variable, and k is the constant of variation. In this article, we will explore the concept of the constant of variation, k, and how to find its value using a given point on the direct variation graph.

What is the Constant of Variation?

The constant of variation, k, is a value that represents the rate at which the dependent variable, y, changes in response to changes in the independent variable, x. In other words, k is a measure of how much y changes when x changes by a certain amount. For example, if y = 2x, then k = 2, which means that for every unit increase in x, y increases by 2 units.

Finding the Constant of Variation

To find the constant of variation, k, we can use the equation y = kx and substitute the given values of x and y. In this case, we are given the point (-3, 2), which means that x = -3 and y = 2. We can substitute these values into the equation y = kx to get:

2 = k(-3)

Solving for k

To solve for k, we can multiply both sides of the equation by -3:

k = 2(-3)

k = -6

However, this is not one of the answer choices. Let's try again.

2 = k(-3)

k = 2/(-3)

k = -2/3

This is one of the answer choices. However, we need to check if it is the correct answer.

Checking the Answer

To check if k = -2/3 is the correct answer, we can substitute this value back into the equation y = kx and see if we get the correct value of y.

y = kx

y = (-2/3)x

y = (-2/3)(-3)

y = 2

Since we get the correct value of y, we can conclude that k = -2/3 is the correct answer.

Conclusion

In conclusion, the constant of variation, k, is a value that represents the rate at which the dependent variable, y, changes in response to changes in the independent variable, x. To find the constant of variation, we can use the equation y = kx and substitute the given values of x and y. In this case, we found that k = -2/3 is the correct answer.

Answer

The correct answer is B. k = -2/3.

Additional Examples

Here are a few more examples of finding the constant of variation:

  • If y = 4x and x = 2, then y = 4(2) = 8. To find k, we can substitute x = 2 and y = 8 into the equation y = kx to get 8 = k(2). Solving for k, we get k = 8/2 = 4.
  • If y = 3x and x = -1, then y = 3(-1) = -3. To find k, we can substitute x = -1 and y = -3 into the equation y = kx to get -3 = k(-1). Solving for k, we get k = -3/(-1) = 3.

Tips and Tricks

Here are a few tips and tricks for finding the constant of variation:

  • Make sure to substitute the given values of x and y into the equation y = kx.
  • Simplify the equation by multiplying both sides by the reciprocal of the coefficient of x.
  • Check your answer by substituting the value of k back into the equation y = kx and seeing if you get the correct value of y.

Conclusion

Q: What is the constant of variation, k?

A: The constant of variation, k, is a value that represents the rate at which the dependent variable, y, changes in response to changes in the independent variable, x.

Q: How do I find the constant of variation, k?

A: To find the constant of variation, k, you can use the equation y = kx and substitute the given values of x and y. Then, solve for k by multiplying both sides of the equation by the reciprocal of the coefficient of x.

Q: What if I have a negative value for k?

A: If you have a negative value for k, it means that the dependent variable, y, decreases as the independent variable, x, increases.

Q: Can I have a zero value for k?

A: Yes, you can have a zero value for k. This means that the dependent variable, y, does not change as the independent variable, x, changes.

Q: Can I have a fractional value for k?

A: Yes, you can have a fractional value for k. This means that the dependent variable, y, changes at a rate that is a fraction of the independent variable, x.

Q: How do I check my answer for k?

A: To check your answer for k, substitute the value of k back into the equation y = kx and see if you get the correct value of y.

Q: What if I get a different answer for k?

A: If you get a different answer for k, it means that your calculation was incorrect. Go back and recheck your work to make sure that you are using the correct values and following the correct steps.

Q: Can I use the constant of variation, k, to solve problems in real-life situations?

A: Yes, you can use the constant of variation, k, to solve problems in real-life situations. For example, if you know that the cost of a product increases at a rate of $2 per unit, you can use the constant of variation, k, to calculate the total cost of the product.

Q: What are some common applications of the constant of variation, k?

A: Some common applications of the constant of variation, k, include:

  • Cost and revenue problems
  • Distance and rate problems
  • Work and time problems
  • Population growth and decay problems

Q: How do I use the constant of variation, k, to solve problems in different fields?

A: To use the constant of variation, k, to solve problems in different fields, you need to understand the context and the variables involved. For example, in economics, the constant of variation, k, can be used to calculate the cost of production, while in physics, it can be used to calculate the distance traveled by an object.

Conclusion

In conclusion, the constant of variation, k, is a powerful tool that can be used to solve problems in a wide range of fields. By understanding the concept of the constant of variation, k, and how to use it to solve problems, you can become a more confident and proficient problem-solver.

Additional Resources

For more information on the constant of variation, k, and how to use it to solve problems, check out the following resources:

  • Khan Academy: Constant of Variation
  • Mathway: Constant of Variation
  • Wolfram Alpha: Constant of Variation

Practice Problems

Try the following practice problems to test your understanding of the constant of variation, k:

  • If y = 2x and x = 4, what is the value of y?
  • If y = 3x and x = -2, what is the value of y?
  • If y = kx and x = 5, what is the value of k if y = 15?

Answer Key

  • If y = 2x and x = 4, then y = 2(4) = 8.
  • If y = 3x and x = -2, then y = 3(-2) = -6.
  • If y = kx and x = 5, then k = 15/5 = 3.