What Is Slope?

How the sign and size speak

Slope is one of the central links between geometry and algebra. It describes the look of the line on the graph and also becomes the coefficient that controls many line equations.

This page keeps the step pattern and the line on the same plane so the number for slope can always be checked against the geometry it is supposed to describe.

A slope of 2 means the line rises 2 units for every 1 unit it runs to the right. A slope of one half rises more gently: it takes a run of 2 to gain a rise of 1. The fraction is not just a number; it is a repeatable movement you can draw on the grid.

• Slope measures rise over run and shows how steep a line is.
• Slope is commonly written as m and calculated as change in y over change in x.
• Parallel nonvertical lines have the same slope, while perpendicular nonvertical lines have slopes that are negative reciprocals.
• A vertical line does not have a defined slope because division by zero is not allowed.

Where slope helps

• Use slope when classifying a line as rising, falling, horizontal, or vertical.
• Use it in line equations, coordinate proofs, and parallel-or-perpendicular line checks.
• Use slope to compare rates of change in algebra and analytic geometry.

Slope checks to make

• Do not reverse the order of subtraction between numerator and denominator; the point order must stay consistent.
• Do not say a vertical line has slope zero; zero slope belongs to horizontal lines.
• Do not confuse the size of the slope with the y-intercept or with line length.