Problem 8 - College Algebra

Question

Find the slope of the line passing through each pair of points or state that the slope is undefined. Then indicate whether the line through the points rises, falls, is horizontal, or is vertical. \((6,-4)\) and \((4,-2)\)

Step-by-Step Solution

Answer

The slope of the line is -1. The line falls.

Step 1: Identify the Coordinates

The points given are \( (6,-4) \) and \( (4,-2) \). So, let \( (x_1, y_1) \) be \( (6,-4) \) and \( (x_2, y_2) \) be \( (4,-2) \).

Step 2: Calculate the Slope

Insert the coordinates into the formula for the slope \( m = \frac{y_2 - y_1}{x_2 - x_1} \). This gives us \( m = \frac{-2 - (-4)}{4 - 6} \). By simplifying the numerator and the denominator separately, the equation becomes \( m = \frac{2}{-2} \), which simplifies to \( m = -1 \).

Step 3: Determine the Direction of the Line

Since the calculated slope is negative, the line falls.

Key Concepts

Coordinate Geometry and Points on a Plane

In coordinate geometry, we identify locations on a two-dimensional plane using points. Each point is represented by a pair of numbers, known as coordinates, typically written as \((x, y)\). The \(x\)-coordinate indicates how far along the horizontal axis the point is, while the \(y\)-coordinate shows how far along the vertical axis it is.
When given two points, such as \((6, -4)\) and \((4, -2)\), the coordinates help us visualize their position on the plane.
Coordinate geometry helps us understand the relationship between points by allowing us to calculate things like distance and position. In this exercise, we focus on finding the slope of the line connecting these points, which tells us about the line's steepness and direction.

Understanding the Slope Formula

The slope of a line is a measure of its steepness and direction. It is symbolized by \(m\). The slope formula, \( m = \frac{y_2 - y_1}{x_2 - x_1} \), is derived from the changes in the \(y\) and \(x\) coordinates of two points. This is often referred to as "rise over run."
In the formula:

• \(y_2 - y_1\) represents the change in the \(y\) coordinates, or the vertical change.
• \(x_2 - x_1\) represents the change in the \(x\) coordinates, or the horizontal change.

By inserting the points \((6, -4)\) and \((4, -2)\) into the formula, the calculation becomes \( m = \frac{-2 - (-4)}{4 - 6} = \frac{2}{-2} = -1\).
The negative result indicates that for every unit increase in the \(x\)-coordinate, the \(y\)-coordinate decreases by 1.

Linear Equations and the Equation of a Line

A linear equation represents a straight line when graphed on a coordinate plane. It is usually written in the form \(y = mx + c\) where:

• \(m\) is the slope.
• \(c\) is the \(y\)-intercept, the point where the line crosses the \(y\)-axis.

Using the slope found from the points \((6, -4)\) and \((4, -2)\), we know \(m = -1\). To find the equation of the line through these points, we choose one point, say \((6, -4)\), and plug it into the equation as follows: \(-4 = -1(6) + c\), solving gives us \(c = 2\). Thus, the equation of this line is \(y = -1x + 2\).
This equation tells us that the line falls as it moves from left to right, consistent with our negative slope.

Understanding Negative Slope

A negative slope indicates a line that decreases or "falls" as it moves from left to right across the graph. This is visualized by its downward tilt. In mathematics, understanding the nature of negative slopes is crucial to interpreting data and trends.
Given a slope of \(-1\), every step (or unit) you move to the right on the \(x\)-axis results in a step (or unit) downward on the \(y\)-axis. Negative slopes often appear in contexts such as economics, where a decrease in one variable might relate to an increase in another.
Identifying whether a line rises or falls helps us understand the relationship between variables and predict future values. In this exercise, the calculated negative slope of \(-1\) tells us the line falls as we move along, providing insight into the axis tendencies and the nature of the relationship between these specific points.