Slope (Rise Over Run)
Slope is the ratio of vertical change (rise) to horizontal change (run) between any two points on a line.
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Slope measures how steep a line is and which direction it goes. A positive slope rises as you move from left to right (uphill); a negative slope falls (downhill). The steeper the line, the larger the absolute value of the slope.
To find slope from two points, count how far you move up or down (the rise), then how far you move left or right (the run), and divide: Slope = Rise ÷ Run. Algebraically, for points (x₁, y₁) and (x₂, y₂), the slope formula is m = (y₂ − y₁) / (x₂ − x₁).
In plain terms: subtract the y-values to get the rise, subtract the x-values in the same order to get the run, then divide rise by run. Slope is constant along the whole line, so any two points give the same answer.
Worked example
Find the slope through (2, 3) and (5, 9).
- Label the points: (x₁, y₁) = (2, 3) and (x₂, y₂) = (5, 9).
- Substitute into the slope formula: m = (9 − 3) / (5 − 2).
- Calculate: 6 / 3 = 2. The line rises 2 units for every 1 unit it moves right.
m = 2
On the test: On the GED & HiSET, slope appears in graphing, tables, and word problems as a steady rate of change.
This concept also comes up in Graphing, The Coordinate Grid.
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