In Exercises 73–76, 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 the line through the points rises, falls, is horizontal, or is vertical. (-a, 0) and (0, -b)

Verified step by step guidance

1. Identify the given points: Point 1 is \((-a, 0)\) and Point 2 is \((0, -b)\).
2. Use the slope formula: \(m = \frac{y_2 - y_1}{x_2 - x_1}\).
3. Substitute the coordinates of the points into the slope formula: \(m = \frac{-b - 0}{0 - (-a)}\).
4. Simplify the expression: \(m = \frac{-b}{a}\).
5. Determine the direction of the line: Since the slope \(m = \frac{-b}{a}\) is negative, the line falls.

Key Concepts

Slope of a Line

The slope of a line measures its steepness and direction, calculated as the change in the y-coordinates divided by the change in the x-coordinates between two points. It is represented by the formula m = (y2 - y1) / (x2 - x1). A positive slope indicates the line rises from left to right, while a negative slope indicates it falls. If the line is vertical, the slope is undefined.

Coordinates of Points

Coordinates are pairs of numbers that define the position of points in a Cartesian plane. Each point is represented as (x, y), where 'x' is the horizontal position and 'y' is the vertical position. In the given question, the points (-a, 0) and (0, -b) indicate specific locations on the plane, which are essential for calculating the slope.

Types of Lines

Lines can be classified based on their slope and orientation. A line is horizontal if it has a slope of 0, meaning it does not rise or fall as it moves along the x-axis. A vertical line has an undefined slope, as it does not change in the x-direction. Understanding these classifications helps in determining the behavior of the line formed by the given points.