Problem 45 - 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: Understand the Slope Formula

The formula to find the slope \( m \) of a line that passes through two points \((x_1, y_1)\) and \((x_2, y_2)\) is \[ m = \frac{y_2 - y_1}{x_2 - x_1} \].

Step 2: Identify the Points

The given points are \((-8, 2)\) and \((-8, 1)\). Thus, \((x_1, y_1) = (-8, 2)\) and \((x_2, y_2) = (-8, 1)\).

Step 3: Calculate the Difference in X-Coordinates

Calculate \( x_2 - x_1 = -8 - (-8) = 0 \).

Step 4: Calculate the Difference in Y-Coordinates

Calculate \( y_2 - y_1 = 1 - 2 = -1 \).

Step 5: Determine if the Slope is Defined

Substitute the differences into the slope formula \( m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{-1}{0} \). Since division by zero is undefined, the slope of the line is undefined.

Key Concepts

Undefined slope

When we talk about the slope of a line being "undefined," it's essential to understand what this indicates about the line itself. A slope is a measure of steepness, and mathematically, it is calculated using a ratio. In the case of two points where the x-coordinates are identical, such as

• (-8, 2) and (-8, 1),

we encounter a vertical line.
Vertical lines have an undefined slope because they go straight up and down and don't "run" horizontally at all. The calculation involves dividing by zero, which is not possible in mathematics. That's why any line with equal x-coordinates at different points will always have an undefined slope, signifying its vertical nature.
Understanding this concept is crucial because it highlights that not all lines have a slope we can calculate normally, and recognizing these cases helps us better comprehend the geometry of lines.

Slope formula

The slope formula is an important tool in coordinate geometry for determining how "tilted" a line is relative to either axis. It's defined using the coordinates of two points on the line,

• extit{(x₁, y₁)} and extit{(x₂, y₂)},

with the formula being expressed as \[ m = \frac{y_2 - y_1}{x_2 - x_1} \].
Here, "m" represents the slope. Simply put, the formula is the change in the y-values divided by the change in the x-values. To better visualize this, think of "rise over run," where "rise" is the difference in y-values (how much the line goes up or down) and "run" is the difference in x-values (how much the line moves left or right).
This straightforward formula provides the basis for understanding line direction and steepness, both crucial aspects of interpreting and analyzing geometric data on a graph.

Coordinate geometry

Coordinate geometry, also known as analytic geometry, is the study of geometric figures using a coordinate system. This field of mathematics allows us to use algebra to gain deeper insights into geometry, with points expressed in terms of their coordinates on a plane.
The beauty of coordinate geometry is in its ability to express the positions of points

• using ordered pairs (x, y),

and then establish relationships between these points, such as determining the slope of a line that passes through them. By using coordinates, problems involving geometry become algebraic, which can be simplified and solved using formulas and equations.
In this system, lines, slopes, and other geometric entities are not just abstract concepts but are tied to specific numbers and values, allowing for precise calculation and visualization of graphs. It's through this lens that students gain a better understanding of spatial relationships, making the seemingly complex world of geometry more approachable.