Problem 32 - College Algebra with Corequisite Support

Question

For the following exercises, find the equation of the line using the given information. The slope is undefined and it passes through the point (2,3) .

Step-by-Step Solution

Answer

Equation of the line: \( x = 2 \).

Step 1: Interpreting the Slope

An undefined slope means the line is vertical. Vertical lines have an equation of the form \( x = a \), where \( a \) is the x-coordinate of any point the line passes through.

Step 2: Using the Given Point

Since the line passes through the point (2,3), we use the x-coordinate from this point to find the equation. Here, the x-coordinate is 2.

Step 3: Writing the Equation

For a vertical line passing through the point (2,3), the equation is \( x = 2 \). This equation indicates a vertical line passing through x = 2 at all points on the y-axis.

Key Concepts

Understanding Undefined Slope

In the world of mathematics, understanding the concept of slope is crucial for analyzing lines on a graph. While we often talk about positive and negative slopes, there's a special case known as an "undefined slope."
When a line has an undefined slope, it means the line is perfectly vertical. In simple terms, a vertical line does not "run" horizontally but only "rises" up and down. Imagine a line like a flagpole or a skyscraper—that's what a vertical line looks like.
This type of line is unique because, for any two points on it, the difference in the x-values (the run) is zero. And since you cannot divide by zero, the slope formula \( m = \frac{\text{rise}}{\text{run}} \) becomes undefined, which is why we call it an "undefined slope."

Characteristics of a Vertical Line

A vertical line has some distinct characteristics that set it apart from other types of lines.

• First and foremost, its slope is undefined, as explored earlier.
• Secondly, all points on a vertical line have the same x-coordinate, while the y-coordinates can be any value—positive, negative, or zero.

This means that if you take any point on the vertical line, you can be certain its x-coordinate will be the same. For example, for the vertical line we are discussing, which passes through the point (2,3), every other point has an x-coordinate of 2, allowing us to write the equation as \( x = 2 \).
From this, we understand that the mathematical expression for a vertical line doesn't depend on y-coordinates, which is different from the slanted or horizontal lines.

Role of the X-Coordinate

The x-coordinate plays a pivotal role in determining the equation of a vertical line. No matter where the line is on the Cartesian plane, its distinguishing feature is that it shares the same x-coordinate at every point along the line.
For instance, if a vertical line passes through the point (2,3), the x-coordinate "2" is key to its equation. The equation \( x = 2 \) tells us that the line intersects the x-axis at 2, and extends up and down without ever moving left or right along the x-axis.
Knowing the x-coordinate is essential because it is the core piece of information needed to write the equation of a vertical line. Unlike other equations, where both x and y play a role, for vertical lines, we only need the x-coordinate to fully capture the essence of its path on a graph.