How to Determine the Relationship Between Lines

Oct 13, 2025 by recoveryshake.com 105 views

Introduction

In mathematics, the relationship between lines is a fundamental concept that helps us understand how different lines interact with each other. Two lines can be either perpendicular, parallel, or neither, depending on their slopes. In this article, we will explore how to determine the relationship between lines based on their slopes and provide a step-by-step guide on how to select the correct answer from each drop-down menu.

What are Perpendicular, Parallel, and Neither Lines?

Before we dive into the details, let's define what perpendicular, parallel, and neither lines are:

Perpendicular lines: Two lines that intersect at a 90-degree angle. Their slopes are negative reciprocals of each other.
Parallel lines: Two lines that lie in the same plane and never intersect. Their slopes are equal.
Neither lines: Two lines that do not intersect and are not parallel. Their slopes are not equal and not negative reciprocals of each other.

How to Determine the Relationship Between Lines

To determine the relationship between lines, we need to find their slopes. The slope of a line is a measure of how steep it is. It can be calculated using the formula:

m = (y2 - y1) / (x2 - x1)

where m is the slope, and (x1, y1) and (x2, y2) are two points on the line.

Finding the Slope of a Line

Let's find the slope of the two lines given in the problem:

{ \begin{array}{l} 6x - 2y = -2 \\ y = 3x + 12 \end{array} \}

For the first line, we can rewrite it in slope-intercept form (y = mx + b), where m is the slope:

6x - 2y = -2 -2y = -6x + 2 y = 3x - 1

So, the slope of the first line is 3.

For the second line, we can see that it is already in slope-intercept form:

y = 3x + 12

So, the slope of the second line is also 3.

Determining the Relationship Between the Lines

Now that we have found the slopes of both lines, we can determine their relationship:

Since the slopes are equal (3 = 3), the lines are parallel.
Since the lines are parallel, they never intersect, and their slopes are not negative reciprocals of each other.

Conclusion

In conclusion, we have learned how to determine the relationship between lines based on their slopes. We have seen that two lines can be either perpendicular, parallel, or neither, depending on their slopes. By finding the slope of each line and comparing them, we can determine their relationship. In this article, we have provided a step-by-step guide on how to select the correct answer from each drop-down menu.

The Square of Their Slopes is

The square of the slopes of the two lines is:

(3)^2 = 9

Therefore, the correct answer is:

Discussion

This problem is a great example of how to apply mathematical concepts to real-world problems. By understanding the relationship between lines, we can solve problems in various fields, such as physics, engineering, and computer science.

Additional Resources

For more information on the relationship between lines, please refer to the following resources:

Khan Academy: Lines and Slopes
Mathway: Lines and Slopes
Wolfram Alpha: Lines and Slopes

Final Thoughts

In conclusion, the relationship between lines is a fundamental concept in mathematics that helps us understand how different lines interact with each other. By finding the slope of each line and comparing them, we can determine their relationship. We hope that this article has provided a clear and concise guide on how to select the correct answer from each drop-down menu.