What is the slope of the line that passes through the points (-8, 6) and (-8, 2)? Write your answer in simplest form.
The slope of a line is a measure of its steepness and direction. Given two points \((x_1, y_1) = (-8, 6)\) and \((x_2, y_2) = (-8, 2)\), we can determine the slope by applying the standard slope formula. This formula expresses the slope \(m\) as the ratio of the change in the vertical coordinate (rise) to the change in the horizontal coordinate (run):
\[m = \frac{y_2 - y_1}{x_2 - x_1}\]
Here, \(x_1 = -8\), \(y_1 = 6\), \(x_2 = -8\), and \(y_2 = 2\).
Now, we substitute the coordinates of our two points into the slope formula:
\[m = \frac{2 - 6}{-8 - (-8)}\]
First, we calculate the change in the vertical coordinates: \(2 - 6 = -4\).
Next, we calculate the change in the horizontal coordinates: \(-8 - (-8) = -8 + 8 = 0\).
Thus, the expression for the slope becomes:
\[m = \frac{-4}{0}\]
In mathematics, division by zero is undefined. This indicates that the line segment connecting these two points has no horizontal change as we move from one point to the other. Consequently, the line is a vertical line. By definition, the slope of any vertical line is undefined.
Understanding the slope of lines is crucial in architecture and engineering for defining the inclination of surfaces. For example, when designing a staircase or a ramp, architects must calculate the slope to ensure safety and accessibility. While most ramps have a specific, measurable slope, a vertical wall represents the extreme case where the slope is undefined, as there is no horizontal distance associated with the height, showing the concept of a vertical orientation in a coordinate system.