Problem 11 - Applied Calculus

Question

Find the slope (if it is defined) of the line determined by each pair of points. $$ (0,-1) \text { and }(4,-1) $$

Step-by-Step Solution

Answer

The slope is 0.

Step 1: Identify the Points

The given points are \((0, -1)\) and \((4, -1)\). Label them as \((x_1, y_1) = (0, -1)\) and \((x_2, y_2) = (4, -1)\).

Step 2: Use the Slope Formula

The formula for the slope \(m\) of a line through two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \].

Step 3: Substitute into the Formula

Insert the values from the points into the slope formula: \[ m = \frac{-1 - (-1)}{4 - 0} \].

Step 4: Simplify the Calculation

Simplify the expression: \[ m = \frac{0}{4} = 0 \]. Since the numerator is zero, the slope of the line is 0.

Key Concepts

Coordinate Geometry

Coordinate geometry, often referred to as analytic geometry, blends algebra and geometry to describe geometric shapes and their properties using coordinates. This field uses a coordinate system to define every point in a plane, most commonly the Cartesian coordinate system.
In this system, each point has an ordered pair of numbers, \(x, y\), representing its position in the plane. The first number, \(x\), is the horizontal distance along the x-axis and the second number, \(y\), is the vertical distance along the y-axis.

• The x-axis is a horizontal line.
• The y-axis is a vertical line.
• The point where the x-axis and y-axis intersect is called the origin with coordinates (0, 0).

This framework allows for the precise description of lines, shapes, and figures within a plane, making tasks like calculating distances, areas, and slopes much more manageable.

Slope Formula

The slope of a line quantifies its steepness and direction. It's an essential concept in coordinate geometry and can be thought of as the measure of how much a line moves up or down for a given horizontal movement.
The slope is determined by the ratio of the difference in the y-coordinates (vertical change) to the difference in the x-coordinates (horizontal change) between two distinct points on the line.
The slope formula is:

• Let \( (x_1, y_1) \) and \( (x_2, y_2) \) be two points on the line.
• The slope, \( m \), is calculated as \[ m = \frac{y_2 - y_1}{x_2 - x_1}. \]

Using the slope formula helps us accurately describe the line's inclination, whether it's going upwards, downwards, or remains constant when observed from left to right.

Horizontal Line

A horizontal line is a straight line where all points on the line have the same y-coordinate, leading to no vertical change regardless of the change in the x-coordinate. Because of this, the slope of any horizontal line is always 0.
This is demonstrated when using the slope formula:

• The differences in y-coordinates are zero (\[-1 - (-1) = 0\]), so the expression for the slope becomes \[\frac{0}{\text{some number}} = 0.\]
• The graph of a horizontal line appears as a straight line parallel to the x-axis.

Understanding this concept allows us to readily identify lines with no incline, simplifying many aspects of coordinate geometry. It's a straightforward indication that the y-values don't change, signifying a constant elevation across the line's length.