Finding the Slope of Line n

Oct 13, 2025 by magicalceylontours.com 125 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 the form y = mx + b, where m is the slope of the line and b is the y-intercept. In this article, we will discuss the equation of line m and the slope of line n, which is perpendicular to line m.

The Equation of Line m

The equation of line m is given as y = (4/3)x + 5. This equation represents a line with a slope of 4/3 and a y-intercept of 5. The slope of a line is a measure of how steep the line is, and it is calculated as the ratio of the vertical change (rise) to the horizontal change (run) between two points on the line.

Perpendicular Lines

Two lines are said to be perpendicular if they intersect at a right angle (90 degrees). In other words, if two lines are perpendicular, then the product of their slopes is equal to -1. This means that if we know the slope of one line, we can easily find the slope of the other line by taking the negative reciprocal of the first slope.

Finding the Slope of Line n

Since line n is perpendicular to line m, we can use the fact that the product of the slopes of two perpendicular lines is equal to -1. Let's call the slope of line n as m'. Then, we have:

m * m' = -1

Substituting the value of m from the equation of line m, we get:

(4/3) * m' = -1

To find the value of m', we can multiply both sides of the equation by the reciprocal of (4/3), which is (3/4). This gives us:

m' = -1 * (3/4) m' = -3/4

Therefore, the slope of line n is -3/4.

Conclusion

In conclusion, the equation of line m is y = (4/3)x + 5, and the slope of line n, which is perpendicular to line m, is -3/4. We used the fact that the product of the slopes of two perpendicular lines is equal to -1 to find the slope of line n.

Example Problems

Here are a few example problems that illustrate the concept of perpendicular lines:

Find the slope of a line that is perpendicular to the line y = 2x + 1.
Find the equation of a line that is perpendicular to the line y = (3/2)x - 2.
Find the slope of a line that is perpendicular to the line y = -x + 4.

Solutions

The slope of a line that is perpendicular to the line y = 2x + 1 is -1/2.
The equation of a line that is perpendicular to the line y = (3/2)x - 2 is y = -2/3x + 4.
The slope of a line that is perpendicular to the line y = -x + 4 is 1.

Applications

The concept of perpendicular lines has many applications in mathematics and real-world problems. Here are a few examples:

Geometry: Perpendicular lines are used to construct right angles and perpendicular bisectors in geometry.
Trigonometry: Perpendicular lines are used to solve problems involving right triangles and trigonometric functions.
Physics: Perpendicular lines are used to describe the motion of objects in physics, such as the trajectory of a projectile.

Real-World Examples

Here are a few real-world examples of perpendicular lines:

Building Design: Architects use perpendicular lines to design buildings and structures that are stable and secure.
Road Design: Engineers use perpendicular lines to design roads and highways that are safe and efficient.
Art: Artists use perpendicular lines to create visually appealing and balanced compositions in art.

Conclusion

Introduction

In our previous article, we discussed the equation of line m and the slope of line n, which is perpendicular to line m. In this article, we will answer some frequently asked questions (FAQs) related to the equation of line m and the slope of line n.

Q: What is the equation of line m?

A: The equation of line m is y = (4/3)x + 5.

Q: What is the slope of line m?

A: The slope of line m is 4/3.

Q: What is the slope of line n?

A: The slope of line n is -3/4.

Q: How do you find the slope of a line that is perpendicular to another line?

A: To find the slope of a line that is perpendicular to another line, you can use the fact that the product of the slopes of two perpendicular lines is equal to -1. Let's call the slope of the first line as m and the slope of the second line as m'. Then, we have:

m * m' = -1

Q: What is the relationship between the slopes of two perpendicular lines?

A: The slopes of two perpendicular lines are negative reciprocals of each other. In other words, if the slope of one line is m, then the slope of the other line is -1/m.

Q: Can you give an example of how to find the slope of a line that is perpendicular to another line?

A: Let's say we have a line with the equation y = 2x + 1. To find the slope of a line that is perpendicular to this line, we can use the fact that the product of the slopes of two perpendicular lines is equal to -1. Let's call the slope of the perpendicular line as m'. Then, we have:

2 * m' = -1

To find the value of m', we can multiply both sides of the equation by the reciprocal of 2, which is 1/2. This gives us:

m' = -1 * (1/2) m' = -1/2

Therefore, the slope of the line that is perpendicular to the line y = 2x + 1 is -1/2.

Q: What are some real-world applications of the concept of perpendicular lines?

A: The concept of perpendicular lines has many real-world applications, including:

Geometry: Perpendicular lines are used to construct right angles and perpendicular bisectors in geometry.
Trigonometry: Perpendicular lines are used to solve problems involving right triangles and trigonometric functions.
Physics: Perpendicular lines are used to describe the motion of objects in physics, such as the trajectory of a projectile.
Building Design: Architects use perpendicular lines to design buildings and structures that are stable and secure.
Road Design: Engineers use perpendicular lines to design roads and highways that are safe and efficient.

Q: Can you give an example of how to use the concept of perpendicular lines in a real-world problem?

A: Let's say we are designing a building and we want to make sure that the walls are perpendicular to each other. We can use the concept of perpendicular lines to ensure that the walls are at a 90-degree angle to each other.

Q: What are some common mistakes to avoid when working with perpendicular lines?

A: Some common mistakes to avoid when working with perpendicular lines include:

Not checking for parallel lines: Make sure to check if the lines are parallel before assuming that they are perpendicular.
Not using the correct formula: Make sure to use the correct formula to find the slope of a line that is perpendicular to another line.
Not considering the context: Make sure to consider the context of the problem and the units of measurement.

Conclusion

In conclusion, the equation of line m is y = (4/3)x + 5, and the slope of line n, which is perpendicular to line m, is -3/4. We answered some frequently asked questions (FAQs) related to the equation of line m and the slope of line n, and provided examples of how to use the concept of perpendicular lines in real-world problems.