What slope actually means
Slope is how steep a line is: how much the line goes up (or down) for every one step you take to the right. A slope of 3 means the line rises 3 units every time you move 1 unit right. A slope of -2 means it falls 2 units for every 1 unit right.
That is the whole idea. Every formula below is just a way of measuring the same thing.
Method 1 — slope from two points
If you know two points on the line, (x₁, y₁) and (x₂, y₂), use the slope formula:
m = (y₂ - y₁) / (x₂ - x₁)
Example: find the slope through (2, 3) and (6, 11). Subtract the y-values: 11 - 3 = 8. Subtract the x-values in the same order: 6 - 2 = 4. So m = 8 / 4 = 2. The line rises 2 units for every 1 unit right.
Example with a negative slope: through (1, 9) and (4, 0). m = (0 - 9) / (4 - 1) = -9 / 3 = -3.
Method 2 — slope from a graph
Pick two points where the line crosses exactly on grid corners — do not estimate. Count how many squares you go up or down (the rise), then how many squares you go right (the run). Slope is rise ÷ run. Going down means the rise is negative.
Method 3 — slope from an equation
Rearrange the equation into slope-intercept form, y = mx + b. Whatever number multiplies x is the slope; b is where the line crosses the y-axis.
Example: 2y = 6x + 10. Divide everything by 2: y = 3x + 5. The slope is 3.
Example: 4x + 2y = 8. Subtract 4x: 2y = -4x + 8. Divide by 2: y = -2x + 4. The slope is -2.
Special cases and common mistakes
A horizontal line has slope 0 — no rise. A vertical line has an undefined slope, because the run is 0 and you cannot divide by 0.
- Subtracting in different orders on top and bottom — always keep the same point first in both.
- Writing run ÷ rise instead of rise ÷ run.
- Dropping the minus sign on a line that goes down left to right.
- Reading the slope off an equation before rearranging it into y = mx + b.
Practice these five
Find the slope through (0, 0) and (5, 15). Through (-2, 4) and (2, -4). Of the line y = 7 - x. Of 3x - y = 6. Of the line through (3, 8) and (3, 1).
Answers: 3, -2, -1, 3, undefined. If any of those surprised you, ask the AI tutor to walk through that exact one.