Problem 5 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 1-6: Sketch the line through \(A\) and \(B\), and find its slope \(m\). $$ A(-3,2), \quad B(-3,5) $$

Step-by-Step Solution

Answer

The line is vertical with an undefined slope.

Step 1: Identifying Coordinates of Points

To sketch the line and find its slope, first identify the coordinates of points \(A\) and \(B\). Here, \(A=(-3, 2)\) and \(B=(-3, 5)\). This means both points have the same \(x\)-coordinate.

Step 2: Understanding Line Orientation

Since points \(A\) and \(B\) have the same \(x\)-coordinate, the line through \(A\) and \(B\) is vertical. Vertical lines do not have a defined slope.

Step 3: Calculating the Slope

The formula for the slope \(m\) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is \(m = \frac{y_2 - y_1}{x_2 - x_1}\). Applying this to points \(A\) and \(B\), we have: \(m = \frac{5 - 2}{-3 - (-3)} = \frac{3}{0}\). Division by zero indicates the slope is undefined.

Step 4: Final Conclusion on Line Type

The slope is undefined because the line is vertical. Consequently, the line through points \(A\) and \(B\) is a vertical line with equation \(x = -3\).

Key Concepts

Understanding Vertical Lines

Vertical lines are unique in the coordinate plane due to their orientation. Unlike most lines which slant to the left or right, vertical lines stand upright, parallel to the y-axis. Such lines are special because every single point along a vertical line shares the same x-coordinate.
For instance, if you have two points such as

• Point A: \( (-3, 2) \)
• Point B: \( (-3, 5) \)

Both points are aligned vertically above each other, reflecting an x-coordinate of -3. As a result, when you draw a line through these points, it’s vertical, and can be described by a simple equation: \( x = -3 \). This equation tells us that no matter how far up or down you go along this line, the x-coordinate remains constant at -3.

Understanding Undefined Slope

The slope of a line is a measure of its steepness, calculated as the change in y-coordinates over the change in x-coordinates between two points. This concept is usually straightforward for most lines. But how do you apply it to vertical lines?
The formula for computing the slope between two points \((x_1, y_1)\) and \((x_2, y_2)\) is\[m = \frac{y_2 - y_1}{x_2 - x_1}\]However, for vertical lines, the x-coordinates are the same:

• \(x_1 = x_2\)

This leads to a denominator of zero:\[m = \frac{3}{0}\]Division by zero is undefined in mathematics. Therefore, the slope of a vertical line cannot be determined using this method, and we say that it has an "undefined slope." Whenever you find yourself in a situation where the x-coordinates of two points are equal, remember: you're dealing with an undefined slope.

Decoding Coordinates of Points

Coordinates are the key to pinpointing the exact position of any point on a two-dimensional plane, like a map. Every point is defined by

• An x-coordinate
• A y-coordinate

Together, these create a pair that resembles \((x, y)\). The x-coordinate tells you how far along—or across—the point is, while the y-coordinate tells you how far up or down it is.
For example, let's look at points \(A\) and \(B\):

• Point A: \((-3, 2)\)
• Point B: \((-3, 5)\)

Here, both points have the same x-coordinate \((-3)\), meaning they lie vertically on the same straight path but differ in their y-coordinates. If points share the same x-coordinate, they will always form a vertical line when connected. In this way, understanding coordinates allows you to sketch lines, figure out their directional nature, and grasp core mathematical properties, like slope.