Identifying the Correct Point

Oct 13, 2025 by windailys.com 125 views

Understanding Direct Variation

Direct variation is a type of relationship between two variables where one variable is a constant multiple of the other. In other words, as one variable increases, the other variable also increases at a constant rate. This relationship can be represented by the equation y = kx, where y is the dependent variable, x is the independent variable, and k is the constant of variation.

Identifying the Correct Point

Given the point (4, -2) is included in a direct variation relationship, we need to find the point that also belongs in this relationship. To do this, we can use the equation y = kx and substitute the given point into the equation.

Step 1: Substitute the Given Point into the Equation

Substituting the point (4, -2) into the equation y = kx, we get:

-2 = k(4)

Step 2: Solve for the Constant of Variation (k)

To solve for k, we can divide both sides of the equation by 4:

k = -2/4 k = -1/2

Step 3: Use the Constant of Variation to Find the Correct Point

Now that we have the value of k, we can use it to find the point that also belongs in the direct variation relationship. We can substitute the value of k into the equation y = kx and try out the different options.

Option A: (-4, 2)

Substituting x = -4 into the equation y = kx, we get:

y = (-1/2)(-4) y = 2

This matches the y-coordinate of the point (-4, 2), so this point also belongs in the direct variation relationship.

Option B: (-4, -2)

Substituting x = -4 into the equation y = kx, we get:

y = (-1/2)(-4) y = 2

This does not match the y-coordinate of the point (-4, -2), so this point does not belong in the direct variation relationship.

Option C: (2, -4)

Substituting x = 2 into the equation y = kx, we get:

y = (-1/2)(2) y = -1

This does not match the y-coordinate of the point (2, -4), so this point does not belong in the direct variation relationship.

Conclusion

Based on the analysis, the point that also belongs in the direct variation relationship is (-4, 2).

Answer

The correct answer is A. (-4, 2).

Discussion

Direct variation is an important concept in mathematics, and understanding how to identify the correct point in a direct variation relationship is crucial for solving problems in algebra and other areas of mathematics. By following the steps outlined in this article, you can confidently identify the correct point in a direct variation relationship and solve problems with ease.

Key Takeaways

Direct variation is a type of relationship between two variables where one variable is a constant multiple of the other.
The equation y = kx represents a direct variation relationship, where y is the dependent variable, x is the independent variable, and k is the constant of variation.
To identify the correct point in a direct variation relationship, substitute the given point into the equation y = kx and solve for the constant of variation (k).
Use the constant of variation to find the point that also belongs in the direct variation relationship by substituting the value of k into the equation y = kx.

Practice Problems

If the point (3, 6) is included in a direct variation relationship, which point also belongs in this relationship?
If the point (2, 4) is included in a direct variation relationship, which point also belongs in this relationship?
If the point (5, 10) is included in a direct variation relationship, which point also belongs in this relationship?

Solutions

The point that also belongs in the direct variation relationship is (6, 18).
The point that also belongs in the direct variation relationship is (4, 8).
The point that also belongs in the direct variation relationship is (10, 20).

Conclusion

Frequently Asked Questions

Q: What is direct variation?

A: Direct variation is a type of relationship between two variables where one variable is a constant multiple of the other. In other words, as one variable increases, the other variable also increases at a constant rate.

Q: How is direct variation represented mathematically?

A: Direct variation is represented mathematically 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 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 identify the correct point in a direct variation relationship?

A: To identify the correct point in a direct variation relationship, substitute the given point into the equation y = kx and solve for the constant of variation (k). Then, use the constant of variation to find the point that also belongs in the direct variation relationship by substituting the value of k into the equation y = kx.

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

A: Some common mistakes to avoid when working with direct variation include:

Failing to identify the correct point in a direct variation relationship
Not using the correct equation (y = kx) to represent the direct variation relationship
Not solving for the constant of variation (k) correctly
Not using the constant of variation to find the point that also belongs in the direct variation relationship

Q: How do I apply direct variation to real-world problems?

A: Direct variation can be applied to a wide range of real-world problems, including:

Modeling population growth
Analyzing the relationship between two variables in a scientific experiment
Understanding the relationship between the cost of a product and its quantity
Predicting the future value of an investment

Q: What are some common applications of direct variation?

A: Some common applications of direct variation include:

Physics: Modeling the motion of objects under constant acceleration
Engineering: Designing systems that require a direct relationship between variables
Economics: Analyzing the relationship between the cost of a product and its quantity
Biology: Modeling population growth and the spread of diseases

Q: How do I determine if a relationship is a direct variation?

A: To determine if a relationship is a direct variation, look for the following characteristics:

A constant rate of change between the variables
A linear relationship between the variables
A consistent ratio between the variables

Q: What are some common misconceptions about direct variation?

A: Some common misconceptions about direct variation include:

Thinking that direct variation only applies to linear relationships
Believing that direct variation only applies to positive values
Assuming that direct variation is only relevant in certain fields, such as physics or engineering

Conclusion

Direct variation is a fundamental concept in mathematics that has numerous applications in real-world problems. By understanding how to identify the correct point in a direct variation relationship and applying direct variation to real-world problems, you can develop a deeper understanding of this important mathematical concept.