Problem 75 - College Algebra

Question

Find the slope of the line passing through each pair of points or state that the slope is undefined. Assume that all variables represent positive real numbers. Then indicate whether either line through the points rises, falls, is horizontal, or is vertical. \((a, b)\) and \((a, b+c)\)

Step-by-Step Solution

Answer

The slope of the line passing through the points \((a, b)\) and \((a, b+c)\) is undefined, indicating that the line is vertical.

Step 1: Use the slope formula

To find the slope of a line passing through two points \((x_1, y_1)\) and \((x_2, y_2)\), the following formula is used: \(\frac{y_2 - y_1}{x_2 - x_1}\). Apply this formula using the given pair of points \((a, b)\) and \((a, b+c)\).

Step 2: Substitute the given values

Substituting these points into the formula, the slope \(m\) of the line is calculated as \(\frac{(b+c) - b}{a - a}\).

Step 3: Simplify the equation

Simplifying the equation, you get \(\frac{c}{0}\).

Step 4: Understand the implications of the results

Since division by zero is undefined in mathematics, the slope of the line is undefined. This means that the line is vertical.

Key Concepts

Undefined Slope

When discussing the slope of a line, an undefined slope occurs when a line is vertical. To understand why, let's recall that the slope of a line shows how steep it is, determined by the change in the vertical direction (rise) over the change in the horizontal direction (run).

In mathematical terms, the slope is calculated using:

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

When you plug in values that cause the denominator \(x_2 - x_1\) to be zero, you end up with division by zero, which is not possible in math.

This is why a line with an undefined slope indicates a vertical line. The rise might be any value, but since the run (horizontal change) is zero, the line shoots straight up or down, connecting the points without moving left or right.

Vertical Line

A vertical line is characterized by moving straight up and down without any horizontal shift. In the coordinate plane, every point on a vertical line shares the same x-coordinate.

This is why, when you have two points like \( (a, b) \) and \( (a, b+c) \), both points have the same x-value, \(a\). The difference is only in the y-coordinates, \(b\) and \(b+c\), which means:

• The line only moves in the vertical direction.

A vertical line is visually distinct from other lines since it doesn't tilt or slope in either a positive or negative direction. It stands perfectly straight, symbolizing a limitless steepness that can't be measured in traditional slope terms.

Slope Formula

The slope formula is a fundamental concept in algebra, used to find the slope of a line through two points. It is expressed as:

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

Here, \(m\) represents the slope, \( (x_1, y_1) \) are the coordinates of the first point, and \( (x_2, y_2) \) are those of the second.

By measuring the difference in the y-coordinates and dividing by the difference in the x-coordinates, you can determine how steep a line is.

In practical terms:

• A positive slope means the line rises.
• A negative slope means it falls.
• Zero slope represents a horizontal line.
• An undefined slope indicates a vertical line.

The slope formula provides a clear way to analyze and understand the direction and steepness of any line on a graph.