Problem 6 - 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. \((4,-1)\) and \((3,-1)\)

Step-by-Step Solution

Answer

The slope of the line passing through the points (4,-1) and (3,-1) is 0. Therefore, the line is horizontal.

Step 1: Find the slope

Determine the slope of the line using the two points (\(4,-1\)) and (\(3,-1\)) and the slope formula which is \((y2 - y1) / (x2 - x1)\). In this case, \(x1 = 4, y1 = -1, x2 = 3, y2 = -1\). Plugging in these values we get \((-1 - -1) / (3 - 4) = 0/(-1) = 0\). The slope of the line through the points is 0.

Step 2: Determine the type of the line

As the slope calculated equals to zero, the line is horizontal, as the rules dictate that when the slope is zero, the line is neither rising nor falling, but is actually a horizontal line.

Key Concepts

Line Equation

When we discuss the equation of a line, we often refer to its slope-intercept form. This form is given as \(y = mx + b\). Here, \(m\) represents the slope, and \(b\) is the y-intercept, the point where the line crosses the y-axis.
The line equation provides a direct relationship between the x and y coordinates of any point on the line. By knowing the slope and y-intercept, you can predict how the line progresses through the coordinate plane.
In any equation of a line, having a slope of 0 (as seen in the original exercise) holds special significance. It indicates that there is no vertical change as you move along the line horizontally. This distinct characteristic leads us directly to the concept of horizontal lines.

Horizontal Line

A horizontal line is one of the most straightforward lines you can encounter in geometry. It extends left and right across the coordinate plane without any inclination. It means all points on the line have the same y-coordinate.
For example, if you look at the line passing through the points \((4,-1)\) and \((3,-1)\), both points have \(y = -1\). This consistency in the y-coordinate tells us that the line is horizontal.
Horizontal lines are also unique because of their slope. They always have a slope of 0. This is because there's no vertical change between any two points along the line. Hence, they neither rise nor fall but remain perfectly flat across the graph.

Slope Formula

The slope of a line is a measurement of its steepness or inclination. It tells us whether a line rises, falls, or stays level as it moves from left to right. The slope is defined as the ratio of the vertical change to the horizontal change between two points on a line.
To find the slope, you can use the formula \(m = \frac{y2 - y1}{x2 - x1}\). Here, \((x1, y1)\) and \((x2, y2)\) represent two distinct points on the line.

• Calculate the change in y (vertical change): \(y2 - y1\).
• Calculate the change in x (horizontal change): \(x2 - x1\).

If the result is a positive number, the line rises. If negative, the line falls. If the result is 0, as in our problem, the line is horizontal. If you cannot define the slope (due to division by zero), the line is vertical.