Problem 8 - Introductory Algebra for College Students

Question

In Exercises \(1-10,\) 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. $$(6,-4) \text { and }(4,-2)$$

Step-by-Step Solution

Answer

The slope of the line passing through the points (6,-4) and (4,-2) is -1. The line through these points is falling.

Step 1: Identifying the coordinates

We have been given two points, \(P_1(6,-4)\) and \(P_2(4,-2)\). These can be identified as \(x_1 = 6\), \(y_1 = -4\), \(x_2 = 4\), and \(y_2 = -2\).

Step 2: Calculating the slope

To find the slope, we apply the formula \(m = (y2 - y1) / (x2 - x1)\). Plugging in our numbers, we get \(m = (-2 - (-4)) / (4 - 6) = 2 / -2 = -1\).

Step 3: Determining line direction

Now that we know the slope is -1, it's clear that the line is falling since the slope is negative. A negative slope indicates a falling line.

Key Concepts

Calculating Slope

When it comes to understanding the basics of slope, it's all about the change in vertical direction (rise) compared to the horizontal direction (run). To calculate the slope, often denoted as \( m \), you'll use the formula \( m = (y_2 - y_1) / (x_2 - x_1) \). This formula essentially measures how steep the line is.
For example, consider the points \( (6, -4) \) and \( (4, -2) \). We calculate the slope as follows:

• First, find the difference in the \( y \)-coordinates: \( -2 - (-4) = 2 \).
• Next, find the difference in the \( x \)-coordinates: \( 4 - 6 = -2 \).
• Finally, divide the difference in \( y \) by the difference in \( x \): \( 2 / -2 = -1 \).

This tells us that the slope is \( -1 \). Calculating the slope this way ensures each line's steepness can be compared quantitatively.

Line Direction

The direction of a line is strongly influenced by its slope. If the slope is:

• Positive, the line rises from left to right.
• Negative, the line falls from left to right.
• Zero, the line is horizontal.
• Undefined, the line is vertical.

Since the slope in our example is \( -1 \), the line falls as you move from left to right. This falling behavior is characteristic of any line with a negative slope.
Understanding the direction helps in visualizing the line's path on a graph.

Undefined Slope

An undefined slope occurs in a vertical line where all points on the line have the same \( x \)-coordinate. In such cases, the change in \( x \) (denominator in the slope formula) is zero, leading to a division by zero, which is undefined.
For example, if you have two points like \( (3, 4) \) and \( (3, -2) \), calculating the slope gives us:

• Difference in \( y \)-coordinates: \( 4 - (-2) = 6 \).
• Difference in \( x \)-coordinates: \( 3 - 3 = 0 \).

Since the denominator becomes zero, the slope is undefined. This confirms a vertical line characteristic.
Remember, whenever the slope is undefined, the line is perfectly vertical.

Falling Line

A falling line is visually identified by its downward trajectory from left to right. This type of line is confirmed through a negative slope, as seen in our example where the slope is \( -1 \).
Falling lines show a decrease in value as you move along the \( x \)-axis, which could represent various real-world scenarios such as a decrease in temperature or stock prices over time.
Grip the concept by envisioning a ski slope or a slide that declines; this idea embodies a falling line graphically. Understanding this type of line helps in interpreting data trends that signify decline.