Find An Equation Of The Line Passing Through The Points { (5, -1)$}$ And { \left(\frac{1}{5}, -1\right)$}$.Sketch The Line.

Introduction

In mathematics, finding the equation of a line passing through two points is a fundamental concept in geometry and algebra. This article will guide you through the process of finding the equation of a line passing through the points {(5, -1)$}$ and {\left(\frac{1}{5}, -1\right)$}$. We will also sketch the line to visualize its position on the coordinate plane.

The Formula for the Equation of a Line

The equation of a line passing through two points ${(x_1, y_1)}$ and ${(x_2, y_2)}$ can be found using the formula:

$$y - y_1 = \frac{y_2 - y_1}{x_2 - x_1} (x - x_1)$$

This formula is derived from the concept of slope, which is the ratio of the vertical change (rise) to the horizontal change (run) between two points on a line.

Finding the Slope

To find the equation of the line passing through the points {(5, -1)$}$ and {\left(\frac{1}{5}, -1\right)$}$, we need to find the slope of the line. The slope can be calculated using the formula:

$$m = \frac{y_2 - y_1}{x_2 - x_1}$$

where ${(x_1, y_1) = (5, -1)}$ and ${(x_2, y_2) = \left(\frac{1}{5}, -1\right)}$.

Plugging in the values, we get:

$$m = \frac{-1 - (-1)}{\frac{1}{5} - 5}$$

$$m = \frac{0}{\frac{1}{5} - 5}$$

$$m = \frac{0}{-\frac{24}{5}}$$

$$m = 0$$

The slope of the line is 0, which means that the line is horizontal.

Finding the Equation of the Line

Now that we have the slope, we can find the equation of the line using the formula:

$$y - y_1 = m (x - x_1)$$

where ${(x_1, y_1) = (5, -1)}$ and ${m = 0}$.

Plugging in the values, we get:

$$y - (-1) = 0 (x - 5)$$

$$y + 1 = 0$$

$$y = -1$$

The equation of the line is ${y = -1}$.

Sketching the Line

To sketch the line, we need to plot the points {(5, -1)$}$ and {\left(\frac{1}{5}, -1\right)$}$ on the coordinate plane. Since the line is horizontal, it will be a straight line parallel to the x-axis.

The line will pass through the points {(5, -1)$}$ and {\left(\frac{1}{5}, -1\right)$}$, and it will be a horizontal line with a y-coordinate of -1.

Conclusion

In this article, we found the equation of the line passing through the points {(5, -1)$}$ and {\left(\frac{1}{5}, -1\right)$}$. We also sketched the line to visualize its position on the coordinate plane. The equation of the line is ${y = -1}$, and it is a horizontal line with a y-coordinate of -1.

Key Takeaways

• The equation of a line passing through two points can be found using the formula ${y - y_1 = \frac{y_2 - y_1}{x_2 - x_1} (x - x_1)}$.
• The slope of a line can be calculated using the formula ${m = \frac{y_2 - y_1}{x_2 - x_1}}$.
• If the slope of a line is 0, the line is horizontal.
• The equation of a horizontal line is of the form ${y = c}$, where ${c}$ is a constant.

Further Reading

• For more information on the equation of a line, see the article on "The Equation of a Line".
• For more information on the slope of a line, see the article on "The Slope of a Line".
• For more information on horizontal lines, see the article on "Horizontal Lines".
  Q&A: Finding the Equation of a Line Passing Through Two Points ===========================================================

Introduction

In our previous article, we discussed how to find the equation of a line passing through two points. In this article, we will answer some frequently asked questions related to this topic.

Q: What is the formula for finding the equation of a line passing through two points?

A: The formula for finding the equation of a line passing through two points ${(x_1, y_1)}$ and ${(x_2, y_2)}$ is:

$$y - y_1 = \frac{y_2 - y_1}{x_2 - x_1} (x - x_1)$$

Q: How do I find the slope of a line passing through two points?

A: To find the slope of a line passing through two points ${(x_1, y_1)}$ and ${(x_2, y_2)}$, you can use the formula:

$$m = \frac{y_2 - y_1}{x_2 - x_1}$$

Q: What if the slope of a line is 0? What does that mean?

A: If the slope of a line is 0, it means that the line is horizontal. A horizontal line has a constant y-coordinate, and its equation is of the form ${y = c}$, where ${c}$ is a constant.

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

A: To find the equation of a horizontal line, you need to find the constant y-coordinate. Since the line is horizontal, the y-coordinate remains the same for all points on the line. Therefore, the equation of a horizontal line is of the form ${y = c}$, where ${c}$ is a constant.

Q: Can I find the equation of a line passing through two points if the points are not on the same horizontal or vertical line?

A: Yes, you can find the equation of a line passing through two points even if the points are not on the same horizontal or vertical line. In this case, you will need to use the formula:

$$y - y_1 = \frac{y_2 - y_1}{x_2 - x_1} (x - x_1)$$

Q: How do I sketch a line passing through two points?

A: To sketch a line passing through two points, you need to plot the points on the coordinate plane and draw a line that passes through both points. If the line is horizontal, it will be a straight line parallel to the x-axis. If the line is vertical, it will be a straight line parallel to the y-axis.

Q: Can I find the equation of a line passing through three points?

A: Yes, you can find the equation of a line passing through three points. However, you will need to use a different method, such as finding the intersection of two lines or using a system of linear equations.

Q: What if I have a system of linear equations? Can I find the equation of a line passing through two points using a system of linear equations?

A: Yes, you can find the equation of a line passing through two points using a system of linear equations. However, this method is more complex and requires a deeper understanding of algebra and geometry.

Conclusion

In this article, we answered some frequently asked questions related to finding the equation of a line passing through two points. We hope that this article has provided you with a better understanding of this topic and has helped you to find the equation of a line passing through two points.

Key Takeaways

• The equation of a line passing through two points can be found using the formula ${y - y_1 = \frac{y_2 - y_1}{x_2 - x_1} (x - x_1)}$.
• The slope of a line can be calculated using the formula ${m = \frac{y_2 - y_1}{x_2 - x_1}}$.
• If the slope of a line is 0, the line is horizontal.
• The equation of a horizontal line is of the form ${y = c}$, where ${c}$ is a constant.
• You can find the equation of a line passing through three points using a different method, such as finding the intersection of two lines or using a system of linear equations.

Further Reading

• For more information on the equation of a line, see the article on "The Equation of a Line".
• For more information on the slope of a line, see the article on "The Slope of a Line".
• For more information on horizontal lines, see the article on "Horizontal Lines".