How is the slope of a line determined when given two points (x₁, y₁) and (x₂, y₂)?

The slope of a line represents the rate of change of the y-coordinate with respect to the x-coordinate and is calculated using two points on the line. When given two points ((x_1, y_1)) and ((x_2, y_2)), the formula for determining the slope (m) is derived from the change in the y-values divided by the change in the x-values, which is captured by the expression ((y_2 - y_1) / (x_2 - x_1)).

This formula indicates that you take the difference in the y-coordinates of the two points (how much the vertical position changes) and divide it by the difference in the x-coordinates (how much the horizontal position changes). This ratio provides a measure of the steepness of the line connecting those points. A positive slope indicates that as you move from left to right along the line, the line rises. Conversely, a negative slope indicates that the line descends.

The other options presented do not correctly represent this relationship. Using the incorrect formulas would yield inaccurate results for the slope, leading to misinterpretations of the line's steepness and direction.