The slope of a line in a two-dimensional coordinate system is a measure of the steepness or incline of the line. It is calculated as the ratio of the vertical change (rise) to the horizontal change (run) between two points on the line.
Mathematically, if you have two points on the line, \((x_1, y_1)\) and \((x_2, y_2)\), the slope \(m\) can be calculated using the formula:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
Here, \(y_2 - y_1\) represents the rise (the change in the y-coordinates), and \(x_2 - x_1\) represents the run (the change in the x-coordinates).
The slope can tell you several things about the line:
- A positive slope indicates that as you move from left to right, the line rises.
- A negative slope indicates that as you move from left to right, the line falls.
- A slope of zero means the line is horizontal.
- An undefined slope (which occurs when \(x_1 = x_2\)) indicates that the line is vertical.
In practical applications, the slope is important in fields such as mathematics, physics, engineering, and economics, as it can represent rates of change, trends, and other relationships between variables.