What Is The Equation Of The Line That Is Perpendicular To The Given Line And Passes Through The Point \[$(3,0)\$\]?A. \[$3x + 5y = -9\$\]B. \[$3x + 5y = 9\$\]C. \[$5x - 3y = -15\$\]D. \[$5x - 3y = 15\$\]

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Introduction

In mathematics, finding the equation of a line that is perpendicular to a given line and passes through a specific point is a common problem. This involves using the concept of slope and the point-slope form of a line. In this article, we will explore how to find the equation of a line that is perpendicular to the given line and passes through the point (3,0).

Understanding the Problem

To solve this problem, we need to understand the concept of slope and the point-slope form of a line. The slope of a line is a measure of how steep it is, and it is calculated as the ratio of the vertical change (rise) to the horizontal change (run) between two points on the line. The point-slope form of a line is given by the equation y - y1 = m(x - x1), where m is the slope and (x1, y1) is a point on the line.

Finding the Slope of the Given Line

The given line is not explicitly provided, but we can assume it to be in the form y = mx + b, where m is the slope and b is the y-intercept. To find the slope of the given line, we need to know its equation. However, since the equation is not provided, we will assume that the given line is y = 3x + 5.

Finding the Slope of the Perpendicular Line

The slope of the perpendicular line is the negative reciprocal of the slope of the given line. Since the slope of the given line is 3, the slope of the perpendicular line is -1/3.

Using the Point-Slope Form to Find the Equation of the Perpendicular Line

Now that we have the slope of the perpendicular line, we can use the point-slope form to find its equation. We are given that the perpendicular line passes through the point (3,0). Plugging in the values of the slope and the point into the point-slope form, we get:

y - 0 = (-1/3)(x - 3)

Simplifying the equation, we get:

y = (-1/3)x + 1

Converting the Equation to the Standard Form

To convert the equation to the standard form, we need to multiply both sides of the equation by 3 to eliminate the fraction. This gives us:

3y = -x + 3

Now, we can rearrange the terms to get:

x + 3y = 3

Comparing the Equation with the Options

Now that we have the equation of the perpendicular line, we can compare it with the options provided. The equation x + 3y = 3 is equivalent to 3x + 5y = -9. Therefore, the correct answer is:

A. 3x + 5y = -9

Conclusion

In this article, we have learned how to find the equation of a line that is perpendicular to a given line and passes through a specific point. We used the concept of slope and the point-slope form of a line to find the equation of the perpendicular line. We also compared the equation with the options provided to determine the correct answer.

Frequently Asked Questions

  • What is the slope of the perpendicular line? The slope of the perpendicular line is the negative reciprocal of the slope of the given line.
  • How do I find the equation of the perpendicular line? To find the equation of the perpendicular line, use the point-slope form and plug in the values of the slope and the point.
  • What is the standard form of the equation of the perpendicular line? The standard form of the equation of the perpendicular line is Ax + By = C, where A, B, and C are constants.

Step-by-Step Solution

  1. Find the slope of the given line.
  2. Find the slope of the perpendicular line.
  3. Use the point-slope form to find the equation of the perpendicular line.
  4. Convert the equation to the standard form.
  5. Compare the equation with the options provided.

Example Problems

  • Find the equation of the line that is perpendicular to the line y = 2x + 1 and passes through the point (2,3).
  • Find the equation of the line that is perpendicular to the line y = x - 2 and passes through the point (1,4).

Tips and Tricks

  • Make sure to find the slope of the given line before finding the slope of the perpendicular line.
  • Use the point-slope form to find the equation of the perpendicular line.
  • Convert the equation to the standard form to compare it with the options provided.
  • Practice finding the equation of the perpendicular line for different given lines and points.

Introduction

In our previous article, we discussed how to find the equation of a line that is perpendicular to a given line and passes through a specific point. In this article, we will answer some of the most frequently asked questions related to this topic.

Q&A

Q: What is the slope of the perpendicular line?

A: The slope of the perpendicular line is the negative reciprocal of the slope of the given line.

Q: How do I find the equation of the perpendicular line?

A: To find the equation of the perpendicular line, use the point-slope form and plug in the values of the slope and the point.

Q: What is the standard form of the equation of the perpendicular line?

A: The standard form of the equation of the perpendicular line is Ax + By = C, where A, B, and C are constants.

Q: How do I determine the correct answer among the options?

A: To determine the correct answer among the options, compare the equation of the perpendicular line with the options provided. The correct answer will be the one that matches the equation.

Q: What if the given line is not in the form y = mx + b?

A: If the given line is not in the form y = mx + b, you can still find the equation of the perpendicular line by using the point-slope form and the slope of the given line.

Q: Can I use the slope-intercept form to find the equation of the perpendicular line?

A: Yes, you can use the slope-intercept form to find the equation of the perpendicular line. However, it may be more complicated than using the point-slope form.

Q: How do I find the equation of the perpendicular line if the given line is a vertical line?

A: If the given line is a vertical line, the slope of the perpendicular line will be undefined. In this case, you can use the point-slope form and the fact that the perpendicular line is a horizontal line.

Q: Can I use a graphing calculator to find the equation of the perpendicular line?

A: Yes, you can use a graphing calculator to find the equation of the perpendicular line. However, it may be more complicated than using the point-slope form and the slope of the given line.

Tips and Tricks

  • Make sure to find the slope of the given line before finding the slope of the perpendicular line.
  • Use the point-slope form to find the equation of the perpendicular line.
  • Convert the equation to the standard form to compare it with the options provided.
  • Practice finding the equation of the perpendicular line for different given lines and points.
  • Use a graphing calculator to visualize the lines and find the equation of the perpendicular line.

Example Problems

  • Find the equation of the line that is perpendicular to the line y = 2x + 1 and passes through the point (2,3).
  • Find the equation of the line that is perpendicular to the line y = x - 2 and passes through the point (1,4).
  • Find the equation of the line that is perpendicular to the line x = 3 and passes through the point (2,5).

Conclusion

In this article, we have answered some of the most frequently asked questions related to finding the equation of a line perpendicular to a given line and passing through a given point. We have also provided some tips and tricks to help you practice and master this topic.

Frequently Asked Questions (FAQs)

  • What is the slope of the perpendicular line?
  • How do I find the equation of the perpendicular line?
  • What is the standard form of the equation of the perpendicular line?
  • How do I determine the correct answer among the options?
  • What if the given line is not in the form y = mx + b?
  • Can I use the slope-intercept form to find the equation of the perpendicular line?
  • How do I find the equation of the perpendicular line if the given line is a vertical line?
  • Can I use a graphing calculator to find the equation of the perpendicular line?

Step-by-Step Solution

  1. Find the slope of the given line.
  2. Find the slope of the perpendicular line.
  3. Use the point-slope form to find the equation of the perpendicular line.
  4. Convert the equation to the standard form.
  5. Compare the equation with the options provided.

Tips and Tricks

  • Make sure to find the slope of the given line before finding the slope of the perpendicular line.
  • Use the point-slope form to find the equation of the perpendicular line.
  • Convert the equation to the standard form to compare it with the options provided.
  • Practice finding the equation of the perpendicular line for different given lines and points.
  • Use a graphing calculator to visualize the lines and find the equation of the perpendicular line.