Problem 19 - College Algebra with Modeling and Visualization

Question

If possible, find the slope of the line passing through each pair of points. $$ (-5,6),(-5,8) $$

Step-by-Step Solution

Answer

The slope is undefined; the line is vertical.

Step 1: Determine the Use of the Slope Formula

To find the slope between two points \((x_1, y_1)\) and \((x_2, y_2)\), we use the slope formula: \(m = \frac{y_2 - y_1}{x_2 - x_1}\). In this problem, the points are \((-5,6)\) and \((-5,8)\).

Step 2: Substitute the Points into the Slope Formula

Substitute the given points into the slope formula: \(m = \frac{8 - 6}{-5 - (-5)}\). This simplifies to \(m = \frac{2}{0}\).

Step 3: Interpret the Result

Since the denominator is zero, the expression \(\frac{2}{0}\) is undefined. This means the slope is undefined, and the line is vertical.

Key Concepts

Slope Formula

The slope formula is a critical tool in understanding how steep a line is on a graph. It helps us determine the direction and angle of the slope between any two given points. The slope formula is represented mathematically as:\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]Here:

• \( y_2 \) and \( y_1 \) are the \( y \)-coordinates of the two points.
• \( x_2 \) and \( x_1 \) are the \( x \)-coordinates of the two points.

The slope \( m \) gives valuable information.

• An increasing line will have a positive slope.
• A decreasing line will have a negative slope.
• If the line is perfectly horizontal, the slope will be zero.

Understanding the slope of a line is fundamental for analyzing linear relationships in algebra and geometry. It can tell us whether relationships are positive or negative or how one variable changes concerning another.

Vertical Line

Vertical lines are unique because they do not have a defined slope like other lines. In a Cartesian plane, a vertical line runs up and down and does not move left or right. This means:

• The \( x \)-coordinate remains constant for all points on the line.
• The line only varies in \( y \)-coordinates.

When calculating the slope of a vertical line using the slope formula, as both \( x_1 \) and \( x_2 \) are the same, the denominator (\( x_2 - x_1 \)) becomes zero.
This results in an undefined situation when substituted into the slope formula.Vertical lines are important in geometry as they represent scenarios where there is no horizontal change, only vertical.

Undefined Slope

An undefined slope occurs when trying to calculate the slope of a vertical line. This is what happens when the denominator of the slope formula becomes zero:\[ m = \frac{y_2 - y_1}{0} \]Division by zero is mathematically undefined, which means the slope of such a line cannot be quantified using standard methods. Here's why an undefined slope occurs:

• A vertical line does not have a change in \( x \)-value—it stays constant.
• Attempting to compute the slope results in dividing by zero.

For real-world applications, understanding that a vertical line implies an undefined slope helps differentiate between differently-directed lines.
This concept is crucial when plotting graphs or interpreting data as it highlights distinct boundaries and constraints within graphical representations.