Slope of Perpendicular Line

Oct 13, 2025 by windailys.com 40 views

Introduction

In mathematics, the concept of perpendicular lines is crucial in understanding various geometric and algebraic relationships. When dealing with lines, it's essential to determine the equation of a line that is perpendicular to a given line and passes through a specific point. In this article, we will explore how to find the equation of a line perpendicular to $y = -3x + 4$ that contains the point $(-6, 2)$ .

Slope-Intercept Form

The slope-intercept form of a line is given by the equation $y = mx + b$ , where $m$ represents the slope of the line and $b$ is the y-intercept. The given line $y = -3x + 4$ is already in slope-intercept form, with a slope of $-3$ and a y-intercept of $4$ . To find the equation of a line perpendicular to this line, we need to determine the slope of the perpendicular line.

Slope of Perpendicular Line

The slope of a line perpendicular to another line is the negative reciprocal of the slope of the original line. In this case, the slope of the given line is $-3$ , so the slope of the perpendicular line is the negative reciprocal of $-3$ , which is $\frac{1}{3}$ .

Equation of Perpendicular Line

Now that we have the slope of the perpendicular line, we can use the point-slope form of a line to find its equation. The point-slope form is given by the equation $y - y_1 = m(x - x_1)$ , where $(x_1, y_1)$ is a point on the line and $m$ is the slope. We are given the point $(-6, 2)$ , so we can substitute these values into the equation along with the slope of the perpendicular line.

Calculating the Equation

Using the point-slope form, we can write the equation of the perpendicular line as:

y - 2 = \frac{1}{3}(x + 6)

To simplify this equation and write it in slope-intercept form, we can distribute the slope to the terms inside the parentheses and then isolate the variable $y$ .

Simplifying the Equation

Distributing the slope to the terms inside the parentheses, we get:

y - 2 = \frac{1}{3}x + 2

Now, we can add $2$ to both sides of the equation to isolate the variable $y$ :

y = \frac{1}{3}x + 4

Conclusion

In this article, we have explored how to find the equation of a line perpendicular to $y = -3x + 4$ that contains the point $(-6, 2)$ . We used the concept of negative reciprocals to determine the slope of the perpendicular line and then used the point-slope form to find its equation. Finally, we simplified the equation and wrote it in slope-intercept form. The equation of the perpendicular line is $y = \frac{1}{3}x + 4$ .

Frequently Asked Questions

What is the slope of a line perpendicular to $y = -3x + 4$ ? The slope of a line perpendicular to $y = -3x + 4$ is the negative reciprocal of the slope of the original line, which is $\frac{1}{3}$ .
How do I find the equation of a line perpendicular to a given line? To find the equation of a line perpendicular to a given line, you need to determine the slope of the perpendicular line using the concept of negative reciprocals and then use the point-slope form to find its equation.
What is the equation of a line perpendicular to $y = -3x + 4$ that contains the point $(-6, 2)$ ? The equation of a line perpendicular to $y = -3x + 4$ that contains the point $(-6, 2)$ is $y = \frac{1}{3}x + 4$ .

Step-by-Step Solution

Determine the slope of the given line.
Find the negative reciprocal of the slope to determine the slope of the perpendicular line.
Use the point-slope form to find the equation of the perpendicular line.
Simplify the equation and write it in slope-intercept form.

Example Problems

Find the equation of a line perpendicular to $y = 2x + 1$ that contains the point $(3, 5)$ .
Find the equation of a line perpendicular to $y = -2x + 3$ that contains the point $(-1, 4)$ .

Real-World Applications

The concept of perpendicular lines has numerous real-world applications in fields such as engineering, architecture, and computer science. For example, in engineering, perpendicular lines are used to design and build structures such as bridges and buildings. In architecture, perpendicular lines are used to create aesthetically pleasing designs and layouts. In computer science, perpendicular lines are used in algorithms and data structures to solve problems and optimize performance.

Conclusion

In conclusion, finding the equation of a line perpendicular to a given line is a crucial concept in mathematics that has numerous real-world applications. By understanding the concept of negative reciprocals and using the point-slope form, we can determine the equation of a line perpendicular to a given line. The equation of a line perpendicular to $y = -3x + 4$ that contains the point $(-6, 2)$ is $y = \frac{1}{3}x + 4$ .

Introduction

In our previous article, we explored how to find the equation of a line perpendicular to a given line. We used the concept of negative reciprocals to determine the slope of the perpendicular line and then used the point-slope form to find its equation. In this article, we will answer some frequently asked questions related to finding the equation of a line perpendicular to a given line.

Q&A

Q: What is the slope of a line perpendicular to $y = -3x + 4$ ?

A: The slope of a line perpendicular to $y = -3x + 4$ is the negative reciprocal of the slope of the original line, which is $\frac{1}{3}$ .

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

A: To find the equation of a line perpendicular to a given line, you need to determine the slope of the perpendicular line using the concept of negative reciprocals and then use the point-slope form to find its equation.

Q: What is the equation of a line perpendicular to $y = -3x + 4$ that contains the point $(-6, 2)$ ?

A: The equation of a line perpendicular to $y = -3x + 4$ that contains the point $(-6, 2)$ is $y = \frac{1}{3}x + 4$ .

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

A: No, you cannot use the slope-intercept form to find the equation of a line perpendicular to a given line. You need to use the point-slope form to find the equation of the perpendicular line.

Q: How do I determine the slope of a line perpendicular to a given line?

A: To determine the slope of a line perpendicular to a given line, you need to find the negative reciprocal of the slope of the original line.

Q: Can I use the point-slope form to find the equation of a line that is not perpendicular to a given line?

A: Yes, you can use the point-slope form to find the equation of a line that is not perpendicular to a given line. However, you need to use the correct slope and point to find the equation of the line.

Q: What is the difference between the slope of a line and the slope of a line perpendicular to it?

A: The slope of a line is the ratio of the vertical change to the horizontal change, while the slope of a line perpendicular to it is the negative reciprocal of the slope of the original line.

Q: Can I use the slope-intercept form to find the equation of a line that is parallel to a given line?

A: No, you cannot use the slope-intercept form to find the equation of a line that is parallel to a given line. You need to use the point-slope form to find the equation of the parallel line.

Q: How do I determine the equation of a line that is parallel to a given line?

A: To determine the equation of a line that is parallel to a given line, you need to use the point-slope form and the correct slope.

Conclusion

In conclusion, finding the equation of a line perpendicular to a given line is a crucial concept in mathematics that has numerous real-world applications. By understanding the concept of negative reciprocals and using the point-slope form, we can determine the equation of a line perpendicular to a given line. We hope that this Q&A article has helped to clarify any questions you may have had about finding the equation of a line perpendicular to a given line.

Frequently Asked Questions

What is the slope of a line perpendicular to $y = -3x + 4$ ?
How do I find the equation of a line perpendicular to a given line?
What is the equation of a line perpendicular to $y = -3x + 4$ that contains the point $(-6, 2)$ ?
Can I use the slope-intercept form to find the equation of a line perpendicular to a given line?
How do I determine the slope of a line perpendicular to a given line?
Can I use the point-slope form to find the equation of a line that is not perpendicular to a given line?
What is the difference between the slope of a line and the slope of a line perpendicular to it?
Can I use the slope-intercept form to find the equation of a line that is parallel to a given line?
How do I determine the equation of a line that is parallel to a given line?

Step-by-Step Solution

Determine the slope of the given line.
Find the negative reciprocal of the slope to determine the slope of the perpendicular line.
Use the point-slope form to find the equation of the perpendicular line.
Simplify the equation and write it in slope-intercept form.

Example Problems

Find the equation of a line perpendicular to $y = 2x + 1$ that contains the point $(3, 5)$ .
Find the equation of a line perpendicular to $y = -2x + 3$ that contains the point $(-1, 4)$ .

Real-World Applications

Conclusion

In conclusion, finding the equation of a line perpendicular to a given line is a crucial concept in mathematics that has numerous real-world applications. By understanding the concept of negative reciprocals and using the point-slope form, we can determine the equation of a line perpendicular to a given line. We hope that this Q&A article has helped to clarify any questions you may have had about finding the equation of a line perpendicular to a given line.