Problem 5 - 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. $$(4,-2)\( and \)(3,-2)$$

Step-by-Step Solution

Answer

The slope of the line passing through the points (4,-2) and (3,-2) is 0, indicating that it is a horizontal line.

Step 1: Identifying the Points

First, identify the two points given in the exercise. In this exercise, we have two points: (4,-2) and (3,-2).

Step 2: Substitute in the Formula

Now substitute the given points in the slope formula, \(m = \frac{y2 - y1}{x2 - x1}\). Where \(x1 = 4\), \(y1 = -2\), \(x2 = 3\) and \(y2 = -2\). This will give us a formula looking like this: \(\frac{-2 - (-2)}{3 - 4}\).

Step 3: Solving the Expression

Now perform the subtraction in the numerator and denominator. \(-2 - (-2) = 0\) and \(3 - 4 = -1\). Thus, our expression becomes \(\frac{0}{-1} = 0\).

Step 4: Identifying the Line

As the slope of the line is 0, this indicates it's a horizontal line. Horizontal lines have a slope of 0

Key Concepts

Algebraic Slope Formula

Understanding the algebraic slope formula is crucial for analyzing the characteristics of lines on a graph. The slope, denoted as m, is a measure of how steep a line is. It's calculated by taking the difference in the y-coordinates (vertical change) and dividing it by the difference in the x-coordinates (horizontal change).

The formula is expressed as:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} \].

When you're given two points, such as \( (x_1, y_1) \) and \( (x_2, y_2) \), you substitute them into the formula to find the slope. If the slope is positive, the line rises as it moves from left to right. If it's negative, the line falls. The absolute value of the slope indicates the steepness—the higher the value, the steeper the line. A very gentle slope might have a value close to zero, while a very steep slope could have a large value, either positive or negative.

Horizontal Lines in Algebra

In algebra, horizontal lines are unique in the sense that they have a slope of 0. This is because there is no vertical change—as you move along the line, the y-coordinate stays the same. Therefore, the formula simplifies because the numerator of the slope equation becomes zero.

The slope of a horizontal line is defined as:
\[ m = \frac{0}{x_2 - x_1} = 0 \].

This says that for any two points on a horizontal line, the y-coordinates are equal and the difference is zero, resulting in a slope of zero. A visual representation would show a line straight across the graph that neither rises nor falls, regardless of how far along the x-axis you go.

Undefined Slope

An undefined slope occurs when you have a vertical line. This situation arises when the x-coordinates of two points on a line are the same; as such, the denominator in the slope formula becomes zero. The formula:
\[ m = \frac{y_2 - y_1}{0} \]

indicates division by zero, which is undefined in mathematics. Hence, the slope is undefined. Vertical lines go straight up and down on the graph and represent situations where for a given value of x there can be many values of y, but for a given y, there's only one corresponding value of x. This characteristic of vertical lines is precisely why their slope cannot be expressed as a numerical value.