Find the slope of the line satisfying the given conditions. See Example 5. horizontal, through (5, 1)

Verified step by step guidance

1. Identify the characteristics of a horizontal line. A horizontal line has a slope of 0 because it does not rise or fall as it moves from left to right.
2. Recognize that the slope of a line is calculated as the change in y-coordinates divided by the change in x-coordinates, often expressed as \( m = \frac{\Delta y}{\Delta x} \).
3. For a horizontal line, the change in y-coordinates (\( \Delta y \)) is 0, since all points on the line have the same y-coordinate.
4. Since \( \Delta y = 0 \), the slope \( m = \frac{0}{\Delta x} = 0 \), regardless of the change in x-coordinates (\( \Delta x \)).
5. Therefore, the slope of the horizontal line passing through the point (5, 1) is 0.

Key Concepts

Slope of a Line

The slope of a line is a measure of its steepness, calculated as the ratio of the vertical change (rise) to the horizontal change (run) between two points on the line. It is often represented by the letter 'm' in the slope-intercept form of a linear equation, y = mx + b. A positive slope indicates the line rises from left to right, while a negative slope indicates it falls.

Horizontal Lines

A horizontal line is a straight line that runs left to right across the coordinate plane and has a slope of 0. This means there is no vertical change as you move along the line, which can be represented by the equation y = b, where b is the y-coordinate of any point on the line. In this case, the line through the point (5, 1) would be y = 1.

Point-Slope Form

The point-slope form of a linear equation is used to express the equation of a line given a point on the line and its slope. It is written as y - y1 = m(x - x1), where (x1, y1) is a point on the line and m is the slope. This form is particularly useful for quickly writing the equation of a line when you know a point and the slope, or in this case, when the slope is known to be 0 for a horizontal line.