Question

Based on material learned earlier in the course. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for the final exam Find the equation of a vertical line passing through the point (5,-3).

Short answer

The equation of the vertical line is \[ x = 5 \]

Step-by-step solution

- Understand Vertical Lines

A vertical line has an undefined slope and goes straight up and down. Therefore, all points on a vertical line have the same x-coordinate.

- Identify the X-Coordinate

Given the point (5, -3), the x-coordinate of this point is 5. All points on the vertical line passing through this point will have this x-coordinate, which is 5.

- Write the Equation

Since the x-coordinate of all points on the vertical line is 5, the equation of the line is simply: \[ x = 5 \]

Key concepts

undefined slope

When discussing vertical lines, one key term to understand is 'undefined slope'. The slope of a line indicates its steepness and can be calculated as the ratio of the vertical change to the horizontal change between two points on the line. For most lines, this gives a finite number.
However, vertical lines are a special case. They rise or fall infinitely without any horizontal change, causing the slope calculation to involve dividing by zero. In mathematics, division by zero is undefined, and hence the slope of a vertical line is considered 'undefined.' Understanding this unique property helps in distinguishing vertical lines from other types of lines.

x-coordinate consistency

Another important feature of vertical lines is 'x-coordinate consistency.' Unlike other lines that slant diagonally or horizontally, vertical lines only move up and down while staying aligned with a constant x-coordinate.
All points on a vertical line share the same x-coordinate. For instance, in the exercise provided, any point on the vertical line passing through (5, -3) will have an x-coordinate of 5. This x-coordinate consistency helps in forming the equation of a vertical line.

To illustrate, let's say you have another vertical line passing through the point (7, 2). Every point on this vertical line will have x = 7, showing how vertical lines maintain x-coordinate consistency.

equation of a vertical line

Once you grasp undefined slope and x-coordinate consistency, writing the equation of a vertical line becomes straightforward. The equation simply reflects the unchanging x-coordinate of all points on the line.
Consider the example from the exercise, where the vertical line passes through (5, -3). Keeping x-coordinate consistency in mind, the equation of the vertical line is x = 5 <|vq_1379|>

This equation signifies that no matter the value of y, the x-coordinate will always be 5.
Another example might be a vertical line passing through (9, 4). The corresponding equation would simply be x = 9.
Recognizing the underlying simplicity of vertical line equations can make solving these types of problems much easier.