What Is The Equation Of The Line That Is Perpendicular To And Has The Same \[$y\$\]-intercept As The Given Line?A. \[$y = \frac{1}{5}x + 1\$\]B. \[$y = \frac{1}{5}x + 5\$\]C. \[$y = 5x + 1\$\]D. \[$y = 5x + 5\$\]

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Introduction

In mathematics, particularly in the field of geometry and algebra, lines play a crucial role in understanding various concepts and theorems. One of the fundamental concepts related to lines is the equation of a line. The equation of a line is a mathematical representation of a line in the form of y = mx + c, where m is the slope of the line and c is the y-intercept. In this article, we will explore the concept of finding the equation of a line that is perpendicular to and has the same y-intercept as the given line.

Understanding the Concept of Perpendicular Lines

Perpendicular lines are lines that intersect at a 90-degree angle. In other words, if two lines are perpendicular, they form a right angle at the point of intersection. The slope of a line is a measure of how steep it is, and the slope of a perpendicular line is the negative reciprocal of the original line's slope. This means that if the slope of the original line is m, the slope of the perpendicular line is -1/m.

Finding the Equation of a Perpendicular Line with the Same y-intercept

To find the equation of a line that is perpendicular to and has the same y-intercept as the given line, we need to follow these steps:

  1. Identify the slope and y-intercept of the given line: The equation of the given line is y = (1/5)x + 1. From this equation, we can see that the slope of the line is 1/5 and the y-intercept is 1.
  2. Find the slope of the perpendicular line: Since the slope of the perpendicular line is the negative reciprocal of the original line's slope, we can find it by taking the negative reciprocal of 1/5, which is -5.
  3. Use the slope and y-intercept to find the equation of the perpendicular line: Now that we have the slope and y-intercept of the perpendicular line, we can use them to find its equation. The equation of a line is y = mx + c, where m is the slope and c is the y-intercept. In this case, the slope is -5 and the y-intercept is 1, so the equation of the perpendicular line is y = -5x + 1.

Comparing the Options

Now that we have found the equation of the perpendicular line, let's compare it with the options given:

  • Option A: y = (1/5)x + 1. This is the equation of the original line, not the perpendicular line.
  • Option B: y = (1/5)x + 5. This is not the equation of the perpendicular line, as it has the same slope and a different y-intercept.
  • Option C: y = 5x + 1. This is not the equation of the perpendicular line, as it has a different slope and the same y-intercept.
  • Option D: y = 5x + 5. This is not the equation of the perpendicular line, as it has a different slope and a different y-intercept.

Conclusion

In conclusion, the equation of the line that is perpendicular to and has the same y-intercept as the given line is y = -5x + 1. This is the only option that satisfies both conditions, making it the correct answer.

Frequently Asked Questions

  • What is the equation of a line? The equation of a line is a mathematical representation of a line in the form of y = mx + c, where m is the slope of the line and c is the y-intercept.
  • What is the slope of a perpendicular line? The slope of a perpendicular line is the negative reciprocal of the original line's slope.
  • How do I find the equation of a perpendicular line with the same y-intercept? To find the equation of a perpendicular line with the same y-intercept, you need to identify the slope and y-intercept of the given line, find the slope of the perpendicular line, and use the slope and y-intercept to find the equation of the perpendicular line.

Final Thoughts

In this article, we explored the concept of finding the equation of a line that is perpendicular to and has the same y-intercept as the given line. We learned how to identify the slope and y-intercept of the given line, find the slope of the perpendicular line, and use the slope and y-intercept to find the equation of the perpendicular line. We also compared the options given and found that the correct answer is y = -5x + 1.

Introduction

In our previous article, we explored the concept of finding the equation of a line that is perpendicular to and has the same y-intercept as the given line. We learned how to identify the slope and y-intercept of the given line, find the slope of the perpendicular line, and use the slope and y-intercept to find the equation of the perpendicular line. In this article, we will answer some frequently asked questions related to this concept.

Q&A

Q: What is the equation of a line?

A: The equation of a line is a mathematical representation of a line in the form of y = mx + c, where m is the slope of the line and c is the y-intercept.

Q: What is the slope of a perpendicular line?

A: The slope of a perpendicular line is the negative reciprocal of the original line's slope.

Q: How do I find the equation of a perpendicular line with the same y-intercept?

A: To find the equation of a perpendicular line with the same y-intercept, you need to:

  1. Identify the slope and y-intercept of the given line.
  2. Find the slope of the perpendicular line by taking the negative reciprocal of the original line's slope.
  3. Use the slope and y-intercept to find the equation of the perpendicular line.

Q: What if the given line has a slope of 0?

A: If the given line has a slope of 0, it means that the line is a horizontal line. In this case, the perpendicular line will have a slope of infinity, and its equation will be x = a, where a is the x-coordinate of the point of intersection.

Q: What if the given line has a slope of infinity?

A: If the given line has a slope of infinity, it means that the line is a vertical line. In this case, the perpendicular line will have a slope of 0, and its equation will be y = b, where b is the y-coordinate of the point of intersection.

Q: Can I find the equation of a perpendicular line with a different y-intercept?

A: Yes, you can find the equation of a perpendicular line with a different y-intercept. However, in this case, the line will not be perpendicular to the original line, but rather parallel to it.

Q: How do I graph a perpendicular line with the same y-intercept?

A: To graph a perpendicular line with the same y-intercept, you need to:

  1. Plot the point of intersection of the two lines.
  2. Draw a line through the point of intersection that is perpendicular to the original line.
  3. Make sure that the line has the same y-intercept as the original line.

Conclusion

In this article, we answered some frequently asked questions related to finding the equation of a perpendicular line with the same y-intercept. We learned how to identify the slope and y-intercept of the given line, find the slope of the perpendicular line, and use the slope and y-intercept to find the equation of the perpendicular line. We also discussed some special cases, such as when the given line has a slope of 0 or infinity, and how to graph a perpendicular line with the same y-intercept.

Final Thoughts

Finding the equation of a perpendicular line with the same y-intercept is an important concept in mathematics, particularly in geometry and algebra. By understanding this concept, you can solve problems related to lines and their equations, and apply it to real-world situations.