A line’s slope compares the change in its vertical coordinate with the corresponding horizontal change. It tells you how much y changes when x increases by one. A positive slope rises from left to right; a negative slope falls. A horizontal line has slope zero, while a vertical line has no defined finite slope.
Divide the vertical change by the horizontal change
Choose two distinct points and subtract their coordinates in the same order. Reversing both subtractions leaves the quotient unchanged; reversing only one changes its sign incorrectly. The horizontal difference must be nonzero.
The angle θ is measured counterclockwise from the positive x-axis. Its tangent gives the slope only when defined. At 90° the line is vertical. An apparent angle in a stretched diagram can be misleading: compare coordinate changes, or use equal axis scales.
Worked example: A(−2, 3) and B(4, 1)
Moving from A to B increases x by 6 and decreases y by 2. Divide −2 by 6 and simplify. The negative result agrees with the descending line.
From a point and slope to the line equation
If the known point has coordinates indexed by zero, every point on the line satisfies the following point-slope equation. It remains valid when m is zero but does not describe a vertical line.
For a line through (2, 3) with slope 2, substitute, expand the brackets and collect the terms. Both final forms describe the same line.
Change the slope and intercept
Move either slider. The line represents y = mx + b; the text states both parameters. A vertical line is outside this finite-slope model.
Slope is not the distance between two points: a ratio describes direction, whereas distance describes separation. When axes carry physical units, slope has vertical units divided by horizontal units.
