Problem 14 - Applied Calculus

Question

Find the slope (if it is defined) of the line determined by each pair of points. $$ (6,-4) \text { and }(6,-3) $$

Step-by-Step Solution

Answer

The slope is not defined because the line is vertical.

Step 1: Recognize the Slope Formula

To find the slope between two points \((x_1, y_1)\) and \((x_2, y_2)\), we use the formula: \( m = \frac{y_2 - y_1}{x_2 - x_1} \). Here, \((x_1, y_1) = (6, -4)\) and \((x_2, y_2) = (6, -3)\).

Step 2: Calculate Differences

Find the differences: \( y_2 - y_1 = -3 - (-4) = -3 + 4 = 1 \), and \( x_2 - x_1 = 6 - 6 = 0 \).

Step 3: Identify the Slope

Substitute these differences into the slope formula: \( m = \frac{1}{0} \). Since division by zero is undefined, the slope is not defined.

Key Concepts

Vertical Lines in Geometry

Vertical lines are unique in geometry. They run straight up and down on a coordinate plane. One of their main characteristics is that all points along a vertical line share the same x-coordinate, which means the line doesn't tilt to the right or left. In our exercise, the line determined by points

• (6, -4)
• (6, -3)

is vertical. Both points have the same x-coordinate, 6. This vertical property significantly impacts the line's slope, which we will discuss next.
Sometimes, you can identify vertical lines simply by looking at the x-coordinates of the points. If the x-values are identical, the line is vertical.

Undefined Slope Explained

The concept of slope helps measure how steep a line is. Slope is often defined as the "rise over run," calculated with the formula:

• \( m = \frac{y_2 - y_1}{x_2 - x_1} \)

A vertical line creates a unique situation where the "run" (or change in x) is zero.
In our case, using points (6, -4) and (6, -3), we calculate:

• \( y_2 - y_1 = 1 \)
• \( x_2 - x_1 = 0 \)
• \( m = \frac{1}{0} \)

Because dividing by zero is not possible in mathematics, the slope of a vertical line is considered "undefined." Rather than indicating a direction, the slope for vertical lines is more about the absence of one along the x-axis.

Coordinate Geometry Basics

Coordinate geometry, also known as analytic geometry, combines algebra and geometry using coordinates. This mathematical field allows us to find precise locations on a plane using ordered pairs, such as (6, -4). In this system:

• The x-coordinate specifies horizontal positioning.
• The y-coordinate specifies vertical positioning.

The understanding of slopes arises naturally in coordinate geometry, as we calculate the direction and steepness of lines connecting points. This exercise effectively demonstrates how two points determine a line's positioning, whether horizontal, vertical, or otherwise. Vertical lines, like our example, show consistency in the x-coordinate, making the overall visual representation simpler to understand.

Mathematical Problem Solving Techniques

Problem-solving in mathematics involves systematic methods. Starting with understanding the problem, we often use known formulas for calculations. In this exercise, we utilized the slope formula to find the result.
Here is a general strategy for solving similar problems:

• Identify the problem clearly. Know what is being asked.
• Select the suitable mathematical tool or formula, such as the slope formula.
• Perform calculations with attention to detail, checking for peculiarities such as zero denominators.
• Interpret the result correctly, placing it in the context of the problem.

These techniques build confidence and efficiency when tackling mathematical questions, ensuring a reliable path to finding solutions.