The Slope of Perpendicular Line s

Oct 13, 2025 by windailys.com 142 views

Introduction

In mathematics, the equation of a line is a fundamental concept that is used to describe the relationship between two variables. The equation of a line can be written in various forms, including the slope-intercept form, which is the most commonly used form. In this article, we will discuss the equation of line r and the slope of perpendicular line s.

The Equation of Line r

The equation of line r is given by:

y = \frac{-3}{8}x + \frac{3}{8}

This equation represents a line with a slope of $\frac{-3}{8}$ and a y-intercept of $\frac{3}{8}$ .

The Slope of Perpendicular Line s

Line s is perpendicular to line r, which means that the slope of line s is the negative reciprocal of the slope of line r. To find the slope of line s, we need to find the negative reciprocal of the slope of line r.

Finding the Negative Reciprocal

The slope of line r is $\frac{-3}{8}$ . To find the negative reciprocal of this slope, we need to multiply it by $-1$ .

\text{Negative reciprocal} = -\frac{1}{\frac{-3}{8}}

To simplify this expression, we can multiply the numerator and denominator by $-1$ .

\text{Negative reciprocal} = -\frac{1}{\frac{-3}{8}} = \frac{1}{\frac{3}{8}} = \frac{8}{3}

Therefore, the slope of line s is $\frac{8}{3}$ .

Conclusion

In this article, we discussed the equation of line r and the slope of perpendicular line s. We found that the slope of line s is the negative reciprocal of the slope of line r, which is $\frac{8}{3}$ . This result is consistent with the definition of perpendicular lines, which states that the slope of a line perpendicular to another line is the negative reciprocal of the slope of the other line.

The Importance of Slope

The slope of a line is an important concept in mathematics that is used to describe the steepness of a line. The slope of a line can be used to determine the direction of the line, as well as the rate at which the line changes. In this article, we saw how the slope of a line can be used to find the slope of a perpendicular line.

Real-World Applications

The concept of slope is used in many real-world applications, including physics, engineering, and economics. For example, the slope of a line can be used to describe the rate at which an object is moving, or the rate at which a quantity is changing. In this article, we saw how the slope of a line can be used to find the slope of a perpendicular line, which is an important concept in many real-world applications.

Final Thoughts

In conclusion, the equation of line r and the slope of perpendicular line s are important concepts in mathematics that are used to describe the relationship between two variables. The slope of a line is an important concept that is used to describe the steepness of a line, and the negative reciprocal of the slope of a line is used to find the slope of a perpendicular line. This article has provided a detailed explanation of the equation of line r and the slope of perpendicular line s, and has highlighted the importance of slope in mathematics and real-world applications.

References

[1] "Algebra" by Michael Artin
[2] "Calculus" by Michael Spivak
[3] "Linear Algebra" by Jim Hefferon

Glossary

Slope: The steepness of a line, which is measured as the ratio of the vertical change to the horizontal change.
Negative reciprocal: The negative reciprocal of a number is the number that, when multiplied by the original number, gives a product of $-1$ .
Perpendicular lines: Two lines that intersect at a right angle, which means that the slope of one line is the negative reciprocal of the slope of the other line.
The Equation of Line r and the Slope of Perpendicular Line s: Q&A ====================================================================

Introduction

In our previous article, we discussed the equation of line r and the slope of perpendicular line s. In this article, we will answer some frequently asked questions about the equation of line r and the slope of perpendicular line s.

Q: What is the equation of line r?

A: The equation of line r is given by:

y = \frac{-3}{8}x + \frac{3}{8}

This equation represents a line with a slope of $\frac{-3}{8}$ and a y-intercept of $\frac{3}{8}$ .

Q: What is the slope of perpendicular line s?

A: The slope of perpendicular line s is the negative reciprocal of the slope of line r. To find the slope of line s, we need to find the negative reciprocal of the slope of line r.

Q: How do I find the negative reciprocal of a slope?

A: To find the negative reciprocal of a slope, we need to multiply it by $-1$ . For example, if the slope of a line is $\frac{a}{b}$ , then the negative reciprocal of the slope is $-\frac{b}{a}$ .

Q: What is the relationship between the slope of a line and the slope of a perpendicular line?

A: The slope of a line and the slope of a perpendicular line are negative reciprocals of each other. This means that if the slope of a line is $\frac{a}{b}$ , then the slope of a perpendicular line is $-\frac{b}{a}$ .

Q: Can you give an example of a line and its perpendicular line?

A: Yes, let's consider a line with the equation $y = 2x + 3$ . The slope of this line is $2$ . To find the slope of a perpendicular line, we need to find the negative reciprocal of the slope of the original line. The negative reciprocal of $2$ is $-\frac{1}{2}$ .

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

A: To find the equation of a perpendicular line, we need to use the slope of the original line and the point of intersection between the two lines. Let's consider a line with the equation $y = 2x + 3$ and a point of intersection $(1, 5)$ . To find the equation of a perpendicular line, we need to use the slope of the original line and the point of intersection.

Q: Can you give an example of finding the equation of a perpendicular line?

A: Yes, let's consider a line with the equation $y = 2x + 3$ and a point of intersection $(1, 5)$ . To find the equation of a perpendicular line, we need to use the slope of the original line and the point of intersection. The slope of the original line is $2$ , and the point of intersection is $(1, 5)$ . To find the equation of a perpendicular line, we need to use the slope of the original line and the point of intersection.

Q: What is the importance of finding the equation of a perpendicular line?

A: Finding the equation of a perpendicular line is important in many real-world applications, including physics, engineering, and economics. For example, in physics, the equation of a perpendicular line can be used to describe the motion of an object. In engineering, the equation of a perpendicular line can be used to design buildings and bridges. In economics, the equation of a perpendicular line can be used to model the behavior of a market.

Conclusion

In this article, we answered some frequently asked questions about the equation of line r and the slope of perpendicular line s. We discussed the equation of line r, the slope of perpendicular line s, and how to find the negative reciprocal of a slope. We also gave examples of finding the equation of a perpendicular line and discussed the importance of finding the equation of a perpendicular line.

References

[1] "Algebra" by Michael Artin
[2] "Calculus" by Michael Spivak
[3] "Linear Algebra" by Jim Hefferon

Glossary

Slope: The steepness of a line, which is measured as the ratio of the vertical change to the horizontal change.
Negative reciprocal: The negative reciprocal of a number is the number that, when multiplied by the original number, gives a product of $-1$ .
Perpendicular lines: Two lines that intersect at a right angle, which means that the slope of one line is the negative reciprocal of the slope of the other line.