The Value Of $y$ Varies Directly With $x$.Solve For $ Y Y Y [/tex] When $x = -8$.Given:${ \begin{array}{c} k = -5 \ y = ? \end{array} }$Remember: $y = Kx$

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The Value of Direct Variation: Solving for y

Understanding Direct Variation

Direct variation is a fundamental concept in mathematics where the value of one variable is directly proportional to the value of another variable. 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 direct variation and solve for y when x = -8.

The Equation of Direct Variation

The equation of direct variation is given by y = kx, where k is the constant of variation. This equation indicates that the value of y is directly proportional to the value of x. The constant of variation, k, determines the rate at which y changes in response to changes in x.

Given Values

In this problem, we are given the value of k, which is -5. We are also given the value of x, which is -8. Our goal is to solve for y when x = -8.

Solving for y

To solve for y, we can substitute the given values into the equation y = kx. Plugging in k = -5 and x = -8, we get:

y = (-5)(-8)

To evaluate this expression, we need to follow the order of operations (PEMDAS):

  1. Multiply -5 and -8:

(-5)(-8) = 40

Therefore, the value of y is 40.

Conclusion

In this article, we explored the concept of direct variation and solved for y when x = -8. We used the equation y = kx and substituted the given values of k and x to find the value of y. The result is y = 40. This problem demonstrates the importance of understanding direct variation and how to apply it to solve real-world problems.

Real-World Applications

Direct variation has numerous real-world applications, including:

  • Physics: The motion of objects can be described using direct variation. For example, the distance traveled by an object is directly proportional to the time it has been traveling.
  • Economics: The price of a product can be described using direct variation. For example, the price of a product is directly proportional to the quantity demanded.
  • Biology: The growth of living organisms can be described using direct variation. For example, the growth of a plant is directly proportional to the amount of sunlight it receives.

Examples and Exercises

Here are some examples and exercises to help you practice solving direct variation problems:

  • Example 1: Solve for y when x = 3 and k = 2.
  • Example 2: Solve for y when x = -2 and k = -3.
  • Exercise 1: Solve for y when x = 4 and k = 5.
  • Exercise 2: Solve for y when x = -6 and k = -2.

Solutions

Here are the solutions to the examples and exercises:

  • Example 1: y = (2)(3) = 6
  • Example 2: y = (-3)(-2) = 6
  • Exercise 1: y = (5)(4) = 20
  • Exercise 2: y = (-2)(-6) = 12

Conclusion

In this article, we explored the concept of direct variation and solved for y when x = -8. We used the equation y = kx and substituted the given values of k and x to find the value of y. The result is y = 40. This problem demonstrates the importance of understanding direct variation and how to apply it to solve real-world problems. We also provided examples and exercises to help you practice solving direct variation problems.
Direct Variation Q&A

Frequently Asked Questions

Direct variation is a fundamental concept in mathematics, and it can be a bit confusing at first. In this article, we will answer some of the most frequently asked questions about direct variation.

Q: What is direct variation?

A: Direct variation is a relationship between two variables where one variable is directly proportional to the other variable. 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.

Q: What is the constant of variation (k)?

A: The constant of variation (k) is a number that determines the rate at which y changes in response to changes in x. It is a measure of the strength of the direct variation relationship.

Q: How do I determine the constant of variation (k)?

A: To determine the constant of variation (k), you need to know the values of y and x for at least two points on the graph. You can then use the equation y = kx to solve for k.

Q: What is the difference between direct variation and inverse variation?

A: Direct variation is a relationship where y is directly proportional to x, while inverse variation is a relationship where y is inversely proportional to x. Inverse variation is represented by the equation y = k/x.

Q: Can direct variation be represented by a graph?

A: Yes, direct variation can be represented by a graph. The graph will be a straight line that passes through the origin (0,0).

Q: How do I graph a direct variation relationship?

A: To graph a direct variation relationship, you need to know the values of y and x for at least two points on the graph. You can then plot these points on a coordinate plane and draw a straight line through them.

Q: Can direct variation be used to model real-world problems?

A: Yes, direct variation can be used to model real-world problems. For example, the distance traveled by an object is directly proportional to the time it has been traveling.

Q: What are some examples of direct variation in real-world problems?

A: Some examples of direct variation in real-world problems include:

  • The price of a product is directly proportional to the quantity demanded.
  • The distance traveled by an object is directly proportional to the time it has been traveling.
  • The growth of a plant is directly proportional to the amount of sunlight it receives.

Q: How do I solve a direct variation problem?

A: To solve a direct variation problem, you need to know the values of y and x for at least two points on the graph. You can then use the equation y = kx to solve for k and find the value of y.

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

A: Some common mistakes to avoid when solving direct variation problems include:

  • Not using the correct equation (y = kx)
  • Not substituting the correct values for y and x
  • Not solving for k correctly

Conclusion

In this article, we answered some of the most frequently asked questions about direct variation. We covered topics such as the definition of direct variation, the constant of variation, and how to graph a direct variation relationship. We also provided examples of direct variation in real-world problems and common mistakes to avoid when solving direct variation problems.