Finding the Slope of the New Path

Oct 13, 2025 by leolavanderia.com.br 71 views

Introduction

When designing a new walking path in a park, it's essential to consider the existing paths and ensure that the new path is safe and accessible for users. In this scenario, we have a walking path represented by the equation $y=-3x-6$ . A new path will be built perpendicular to this path, and the two paths will intersect at the point $(-3,3)$ . Our goal is to find the equation that represents the new path.

Understanding the Problem

To find the equation of the new path, we need to understand the concept of perpendicular lines. Two lines are perpendicular if their slopes are negative reciprocals of each other. In other words, if the slope of one line is $m$ , then the slope of the perpendicular line is $-\frac{1}{m}$ .

Finding the Slope of the New Path

The slope of the existing path is $-3$ , which means that the slope of the new path will be the negative reciprocal of $-3$ . To find the negative reciprocal of $-3$ , we can use the following formula:

-\frac{1}{-3} = \frac{1}{3}

Therefore, the slope of the new path is $\frac{1}{3}$ .

Writing the Equation of the New Path

Now that we have the slope of the new path, we can write its equation in the form $y = mx + b$ , where $m$ is the slope and $b$ is the y-intercept. Since the new path passes through the point $(-3,3)$ , we can substitute these values into the equation to find the value of $b$ .

3 = \frac{1}{3}(-3) + b

Simplifying the equation, we get:

3 = -1 + b

b = 4

Therefore, the equation of the new path is:

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

Conclusion

In this article, we found the equation of a new walking path that is perpendicular to an existing path in a park. We used the concept of negative reciprocals to find the slope of the new path and then wrote its equation in the form $y = mx + b$ . The new path passes through the point $(-3,3)$ and has a slope of $\frac{1}{3}$ . The equation of the new path is $y = \frac{1}{3}x + 4$ .

Key Takeaways

The slope of the new path is the negative reciprocal of the slope of the existing path.
The equation of the new path can be written in the form $y = mx + b$ , where $m$ is the slope and $b$ is the y-intercept.
The new path passes through the point $(-3,3)$ and has a slope of $\frac{1}{3}$ .

Real-World Applications

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

Designing buildings and bridges
Creating road networks and traffic patterns
Developing architectural designs and blueprints
Analyzing data and creating visualizations

Future Directions

In the future, we can explore more complex scenarios, such as finding the equation of a new path that is perpendicular to multiple existing paths. We can also investigate the use of technology, such as computer-aided design (CAD) software, to design and visualize new paths.

References

[1] Khan Academy. (n.d.). Slope and Perpendicular Lines. Retrieved from <https://www.khanacademy.org/math/algebra/x2f7f7c/x2f7f7d/x2f7f7e/x2f7f7f/x2f7f7g/x2f7f7h/x2f7f7i/x2f7f7j/x2f7f7k/x2f7f7l/x2f7f7m/x2f7f7n/x2f7f7o/x2f7f7p/x2f7f7q/x2f7f7r/x2f7f7s/x2f7f7t/x2f7f7u/x2f7f7v/x2f7f7w/x2f7f7x/x2f7f7y/x2f7f7z/x2f7f7aa/x2f7f7ab/x2f7f7ac/x2f7f7ad/x2f7f7ae/x2f7f7af/x2f7f7ag/x2f7f7ah/x2f7f7ai/x2f7f7aj/x2f7f7ak/x2f7f7al/x2f7f7am/x2f7f7an/x2f7f7ao/x2f7f7ap/x2f7f7aq/x2f7f7ar/x2f7f7as/x2f7f7at/x2f7f7au/x2f7f7av/x2f7f7aw/x2f7f7ax/x2f7f7ay/x2f7f7az/x2f7f7ba/x2f7f7bb/x2f7f7bc/x2f7f7bd/x2f7f7be/x2f7f7bf/x2f7f7bg/x2f7f7bh/x2f7f7bi/x2f7f7bj/x2f7f7bk/x2f7f7bl/x2f7f7bm/x2f7f7bn/x2f7f7bo/x2f7f7bp/x2f7f7bq/x2f7f7br/x2f7f7bs/x2f7f7bt/x2f7f7bu/x2f7f7bv/x2f7f7bw/x2f7f7bx/x2f7f7by/x2f7f7bz/x2f7f7ca/x2f7f7cb/x2f7f7cc/x2f7f7cd/x2f7f7ce/x2f7f7cf/x2f7f7cg/x2f7f7ch/x2f7f7ci/x2f7f7cj/x2f7f7ck/x2f7f7cl/x2f7f7cm/x2f7f7cn/x2f7f7co/x2f7f7cp/x2f7f7cq/x2f7f7cr/x2f7f7cs/x2f7f7ct/x2f7f7cu/x2f7f7cv/x2f7f7cw/x2f7f7cx/x2f7f7cy/x2f7f7cz/x2f7f7da/x2f7f7db/x2f7f7dc/x2f7f7dd/x2f7f7de/x2f7f7df/x2f7f7dg/x2f7f7dh/x2f7f7di/x2f7f7dj/x2f7f7dk/x2f7f7dl/x2f7f7dm/x2f7f7dn/x2f7f7do/x2f7f7dp/x2f7f7dq/x2f7f7dr/x2f7f7ds/x2f7f7dt/x2f7f7du/x2f7f7dv/x2f7f7dw/x2f7f7dx/x2f7f7dy/x2f7f7dz/x2f7f7ea/x2f7f7eb/x2f7f7ec/x2f7f7ed/x2f7f7ee/x2f7f7ef/x2f7f7eg/x2f7f7eh/x2f7f7ei/x2f7f7ej/x2f7f7ek/x2f7f7el/x2f7f7em/x2f7f7en/x2f7f7eo/x2f7f7ep/x2f7f7eq/x2f7f7er/x2f7f7es/x2f7f7et/x2f7f7eu/x2f7f7ev/x2f7f7ew/x2f7f7ex/x2f7f7ey/x2f7f7ez/x2f7f7fa/x2f7f7fb/x2f7f7fc/x2f7f7fd/x2f7f7fe/x2f7f7ff/x2f7f7fg/x2f7f7fh/x2f7f7fi/x2f7f7fj/x2f7f7fk/x2f7f7fl/x2f7f7fm/x2f7f7fn/x2f7f7fo/x2f7f7fp/x2f7f7fq/x2f7f7fr/x2f7f7fs/x2f7f7ft/x2f7f7fu/x2f7f7fv/x2f7f7fw/x2f7f7fx/x2f7f7fy/x2f7f7fz/x2f7f7

Introduction

In our previous article, we explored the concept of finding the equation of a new walking path that is perpendicular to an existing path in a park. We used the concept of negative reciprocals to find the slope of the new path and then wrote its equation in the form $y = mx + b$ . In this article, we will answer some frequently asked questions related to this topic.

Q&A

Q: What is the concept of negative reciprocals?

A: The concept of negative reciprocals refers to the relationship between the slopes of two perpendicular lines. If the slope of one line is $m$ , then the slope of the perpendicular line is $-\frac{1}{m}$ .

Q: How do I find the equation of a new path that is perpendicular to an existing path?

A: To find the equation of a new path that is perpendicular to an existing path, you need to find the slope of the new path, which is the negative reciprocal of the slope of the existing path. Then, you can write the equation of the new path in the form $y = mx + b$ , where $m$ is the slope and $b$ is the y-intercept.

Q: What is the significance of the point of intersection?

A: The point of intersection is the point where the two paths meet. In this case, the point of intersection is $(-3,3)$ .

Q: How do I find the equation of a new path that passes through a given point?

A: To find the equation of a new path that passes through a given point, you need to substitute the coordinates of the point into the equation of the line. For example, if the equation of the line is $y = mx + b$ and the point is $(x_1, y_1)$ , then you can substitute $x_1$ and $y_1$ into the equation to find the value of $b$ .

Q: What are some real-world applications of finding the equation of a new path?

A: Finding the equation of a new path has many real-world applications, including designing buildings and bridges, creating road networks and traffic patterns, developing architectural designs and blueprints, and analyzing data and creating visualizations.

Q: Can I use technology to find the equation of a new path?

A: Yes, you can use technology, such as computer-aided design (CAD) software, to find the equation of a new path. CAD software can help you visualize and design the path, and can also provide you with the equation of the path.

Conclusion

In this article, we answered some frequently asked questions related to finding the equation of a new walking path that is perpendicular to an existing path in a park. We hope that this article has provided you with a better understanding of the concept of negative reciprocals and how to find the equation of a new path.

Key Takeaways

The concept of negative reciprocals refers to the relationship between the slopes of two perpendicular lines.
To find the equation of a new path that is perpendicular to an existing path, you need to find the slope of the new path, which is the negative reciprocal of the slope of the existing path.
The point of intersection is the point where the two paths meet.
Finding the equation of a new path has many real-world applications, including designing buildings and bridges, creating road networks and traffic patterns, developing architectural designs and blueprints, and analyzing data and creating visualizations.

Future Directions

References

[1] Khan Academy. (n.d.). Slope and Perpendicular Lines. Retrieved from <https://www.khanacademy.org/math/algebra/x2f7f7c/x2f7f7d/x2f7f7e/x2f7f7f/x2f7f7g/x2f7f7h/x2f7f7i/x2f7f7j/x2f7f7k/x2f7f7l/x2f7f7m/x2f7f7n/x2f7f7o/x2f7f7p/x2f7f7q/x2f7f7r/x2f7f7s/x2f7f7t/x2f7f7u/x2f7f7v/x2f7f7w/x2f7f7x/x2f7f7y/x2f7f7z/x2f7f7aa/x2f7f7ab/x2f7f7ac/x2f7f7ad/x2f7f7ae/x2f7f7af/x2f7f7ag/x2f7f7ah/x2f7f7ai/x2f7f7aj/x2f7f7ak/x2f7f7al/x2f7f7am/x2f7f7an/x2f7f7ao/x2f7f7ap/x2f7f7aq/x2f7f7ar/x2f7f7as/x2f7f7at/x2f7f7au/x2f7f7av/x2f7f7aw/x2f7f7ax/x2f7f7ay/x2f7f7az/x2f7f7ba/x2f7f7bb/x2f7f7bc/x2f7f7bd/x2f7f7be/x2f7f7bf/x2f7f7bg/x2f7f7bh/x2f7f7bi/x2f7f7bj/x2f7f7bk/x2f7f7bl/x2f7f7bm/x2f7f7bn/x2f7f7bo/x2f7f7bp/x2f7f7bq/x2f7f7br/x2f7f7bs/x2f7f7bt/x2f7f7bu/x2f7f7bv/x2f7f7bw/x2f7f7bx/x2f7f7by/x2f7f7bz/x2f7f7ca/x2f7f7cb/x2f7f7cc/x2f7f7cd/x2f7f7ce/x2f7f7cf/x2f7f7cg/x2f7f7ch/x2f7f7ci/x2f7f7cj/x2f7f7ck/x2f7f7cl/x2f7f7cm/x2f7f7cn/x2f7f7co/x2f7f7cp/x2f7f7cq/x2f7f7cr/x2f7f7cs/x2f7f7ct/x2f7f7cu/x2f7f7cv/x2f7f7cw/x2f7f7cx/x2f7f7cy/x2f7f7cz/x2f7f7da/x2f7f7db/x2f7f7dc/x2f7f7dd/x2f7f7de/x2f7f7df/x2f7f7dg/x2f7f7dh/x2f7f7di/x2f7f7dj/x2f7f7dk/x2f7f7dl/x2f7f7dm/x2f7f7dn/x2f7f7do/x2f7f7dp/x2f7f7dq/x2f7f7dr/x2f7f7ds/x2f7f7dt/x2f7f7du/x2f7f7dv/x2f7f7dw/x2f7f7dx/x2f7f7dy/x2f7f7dz/x2f7f7ea/x2f7f7eb/x2f7f7ec/x2f7f7ed/x2f7f7ee/x2f7f7ef/x2f7f7eg/x2f7f7eh/x2f7f7ei/x2f7f7ej/x2f7f7ek/x2f7f7el/x2f7f7em/x2f7f7en/x2f7f7eo/x2f7f7ep/x2f7f7eq/x2f7f7er/x2f7f7es/x2f7f7et/x2f7f7eu/x2f7f7ev/x2f7f7ew/x2f7f7ex/x2f7f7ey/x2f7f7ez/x2f7f7fa/x2f7f7fb/x2f7f7fc/x2f7f7fd/x2f7f7fe/x2f7f7ff/x2f7f7fg/x2f7f7fh/x2f7f7fi/x2f7f7fj/x2f7f7fk/x2f7f7fl/x2f7f7fm/x2f7f7fn/x2f7f7fo/x2f7f7fp/x2f7f7fq/x2f7f7fr/x2f7f7fs/x2f7f7