Problem 13 - Brief Applied Calculus

Question

Find the slope (if it is defined) of the line determined by each pair of points. \((2,-1)\) and \((2,5)\)

Step-by-Step Solution

Answer

The slope is undefined because the line is vertical.

Step 1: Identify the Variables

In the given points, the first point is \((x_1, y_1) = (2, -1)\) and the second point is \((x_2, y_2) = (2, 5)\). These values will be used to calculate the slope.

Step 2: Apply the Slope Formula

The formula for the slope, \(m\), between two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by:\[m = \frac{y_2 - y_1}{x_2 - x_1}\]Substitute the given values: \(y_2 = 5\), \(y_1 = -1\), \(x_2 = 2\), and \(x_1 = 2\).

Step 3: Calculate the Numerator

The numerator is calculated as \(y_2 - y_1 = 5 - (-1) = 5 + 1 = 6\).

Step 4: Calculate the Denominator

The denominator is calculated as \(x_2 - x_1 = 2 - 2 = 0\).

Step 5: Determining the Slope

Since the denominator is 0, the slope \(m\) is undefined because division by zero is not possible. The line is vertical.

Key Concepts

Vertical Lines

A vertical line is a unique concept in geometry. Unlike most lines that run diagonally or horizontally, vertical lines go straight up and down. They are characterized by having the same x-coordinate for every point on the line. Let's take the points

• \((2, -1)\)
• \((2, 5)\)

Here, both points have the x-coordinate of 2, which means every point on this line has x = 2. This specific x-value stays consistent, making the line vertical.

Vertical lines behave differently in terms of slope compared to other lines. You won't find a 'rise over run' like in more common lines. Instead, the 'run' or horizontal change is zero, which takes us to our next concept, the undefined slope.

Undefined Slope

Slope usually measures how steep a line is. It is usually calculated as \[m = \frac{y_2 - y_1}{x_2 - x_1}\]
This formula means you take the difference in y-values and divide it by the difference in x-values. This gives the change in vertical position per unit of horizontal change.

• However, when dealing with a vertical line, the formula behaves differently.
• For the line determined by the points \((2, -1)\)and \((2,5)\)\(x_2 - x_1 = 0\).

This results in division by zero, which is undefined in mathematics.
Thus, we say that a vertical line has an undefined slope. This undefined nature is important because it signifies that vertical lines stand apart uniquely in geometry.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, explores the connection between algebra and geometry through graphs and coordinates. It is usually done in the Cartesian plane, where you plot x and y coordinates to create shapes or lines.

In our example, the two points

• \((2, -1)\)
• \((2,5)\)

are located along the y-axis at x = 2.
This is a perfect example of how coordinate geometry helps. It helps visualize and understand the spatial relationship of points. It clarifies why certain lines, like vertical ones, behave the way they do.
Understanding the basics of coordinate geometry equips you with tools to solve problems in geometry using algebra. It breaks down complex geometric ideas into simple, computational procedures, using coordinates as the bridge between two realms of math.