What is the definition of slope in the context of a line?

The definition of slope in the context of a line is indeed the ratio of vertical change to horizontal change. This concept is fundamental in understanding linear relationships in mathematics, particularly in algebra and geometry.

Slope is denoted as (m) and is calculated using the formula (m = \frac{y_2 - y_1}{x_2 - x_1}), where ((x_1, y_1)) and ((x_2, y_2)) are two distinct points on the line. The numerator (y_2 - y_1) represents the vertical change (how much the y-values change between the two points), while the denominator (x_2 - x_1) represents the horizontal change (how much the x-values change). Thus, slope quantifies how steeply a line rises or falls as you move from left to right across the graph.

This ratio provides valuable information about the direction and steepness of the line—if the slope is positive, the line rises; if negative, it falls; and if zero, the line is horizontal. Understanding slope is crucial for interpreting graphs in various fields, including physics, economics, and engineering.