Understanding Perpendicular and Parallel Lines to the x-axis

In this article, we will explore the equations of lines that are either perpendicular or parallel to the x-axis. Specifically, we will analyze a line that passes through a given point and determine which conditions are met for perpendicularity or parallelity.

Perpendicular Line to the x-axis

Firstly, let's consider the equation of a line that is perpendicular to the x-axis and passes through a specific point, such as P2, -3.

Since a line perpendicular to the x-axis is a vertical line, the x-coordinate for all points on this line remains constant. Therefore, the equation of such a line is:

x 2

This equation ensures that every point on the line has an x-coordinate of 2, regardless of the y-coordinate.

For a more in-depth breakdown:

The x-coordinate of the line is the same as the x-coordinate of the point through which the line passes (2 in this case). Any vertical line can be represented as x k, where k is a constant (the x-coordinate of the line). Thus, the equation x 2 represents all points that are vertically aligned with the point (2, -3).

Parallel Line to the x-axis

Next, we will consider a line that is parallel to the x-axis and passes through the same point, P2, -3.

Unlike a vertical line, a horizontal line (parallel to the x-axis) has a constant y-coordinate. Therefore, the equation of this line is:

y -3

This can also be derived from the general slope-intercept form, y mx b, where:

The slope (m) of a horizontal line is 0. Given the point (2, -3), the y-coordinate is -3, so b -3. Thus, the equation is y 0*x -3, which simplifies to y -3.

Further Clarification of Line Equations

Some may be confused as to why y -3 represents a horizontal line. Let's clear up the confusion:

The equation x 2 represents a vertical line, and any point on this line will have an x-coordinate of 2.

Similarly, y -3 represents a horizontal line, and any point on this line will have a y-coordinate of -3, regardless of the x-coordinate.

It's important to differentiate between the two:

A vertical line (x k) is parallel to the y-axis and changes its y-coordinate but not its x-coordinate. A horizontal line (y k) is parallel to the x-axis and changes its x-coordinate but not its y-coordinate.

For example, the line y -3 is parallel to the x-axis and is not the same as the line x -3, which is a vertical line.

Conclusion

In summary, the equation of a line that is perpendicular to the x-axis and passes through a point (2, -3) is:

x 2

For a line parallel to the x-axis and passing through the same point, the equation is: