Problem 44 - A Graphical Approach to College Algebra

Question

Find the slope (if defined) of the line that passes through the given points. $$44 .(-8,2) \text { and }(-8,1)$$

Step-by-Step Solution

Answer

The slope is undefined because the line is vertical.

Step 1: Identify the Points

We have the points \((-8, 2)\) and \((-8, 1)\). These points lie on the line we need to analyze.

Step 2: Check for Vertical Line

Notice that the x-coordinates of both points are the same, \(x = -8\). This indicates that the line is vertical.

Step 3: Understand Vertical Line Slope

For a vertical line, the slope is undefined. Slope is calculated using the change in \( y \) over the change in \( x \), but in this case, \( \Delta x = 0 \).

Step 4: Calculate the Slope (or Confirm Undefined)

Attempt to calculate the slope using the formula: \( m = \frac{y_2 - y_1}{x_2 - x_1} \).Substituting our known values, \( m = \frac{1 - 2}{-8 - (-8)} = \frac{-1}{0} \). Division by zero confirms the slope is undefined.

Key Concepts

Vertical Line

A vertical line is a type of line in coordinate geometry where all points on the line share the same x-coordinate. This means, for any two points on a vertical line, say

• Point A \((x_1, y_1)\)
• Point B \((x_2, y_2)\)

the x-values will be identical (i.e., \((x_1 = x_2)\)).
This property of having a constant x-coordinate makes vertical lines unique compared to horizontal or slanted lines, which have variations in both x and y coordinates as you move along them. Vertical lines run from top to bottom on a graph, standing perpendicular to the horizontal axis. This distinct position gives such lines their characteristic appearance in geometry.

Undefined Slope

The concept of slope in coordinate geometry usually refers to the steepness of a line, calculated as the ratio of the vertical change (change in y) over the horizontal change (change in x) between two points. The formula for slope is \[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
However, in the case of a vertical line, the x-coordinates of the points are the same. This results in \(x_2 - x_1 = 0\). Since division by zero is mathematically undefined, the slope of a vertical line is considered undefined.

• An undefined slope indicates that the line doesn't have a well-defined angle with the horizontal.
• All vertical lines have slopes that are declared as undefined.

Thus, an undefined slope highlights the unique and specific nature of vertical lines in the context of linear equations.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, involves studying geometrical shapes through a coordinate system. Using points defined by pairs \(x, y\) in the Cartesian coordinate system, it helps to visually and algebraically examine a line or a figure.
This approach allows you to precisely describe and manipulate the properties of lines, circles, and various other shapes using equations.

• The x-coordinate represents the horizontal displacement of a point from the origin.
• The y-coordinate represents the vertical displacement.

For the line passing through \((-8, 2)\) and \((-8, 1)\), coordinate geometry enables the recognition of vertical alignment due to the identical x-values. This powerful method not only illuminates the unique properties of shapes and lines including vertical lines, but also supports solving complex geometry problems using algebraic techniques.