Slope in simple words
The slope of a line tells you how steep it is. It is the rise (how far the line goes up) divided by the run (how far it goes to the right). A bigger slope means a steeper line. A negative slope means the line goes down as you move right.
To find the slope from two points, take away the y values to get the rise and take away the x values to get the run, then divide: m = (y₂ − y₁) ÷ (x₂ − x₁).
Example: points (1, 2) and (5, 10). The rise is 10 − 2 = 8 and the run is 5 − 1 = 4, so the slope is 8 ÷ 4 = 2. The line goes up 2 for every 1 step to the right.
A flat line has a slope of 0. A straight up-and-down line has no slope number at all, because you would have to divide by zero.
Formula and method
Slope m = (y₂−y₁)/(x₂−x₁), intercept b = y₁−mx₁ and the line is y = mx+b. For two points, distance is √((x₂−x₁)²+(y₂−y₁)²), and midpoint averages each coordinate.
Worked example
Points (1,2) and (5,10) give slope 8/4 = 2, line y = 2x and midpoint (3,6). Their distance is √80 ≈ 8.94427191; a perpendicular line has slope −1/2.
Inputs and limitations
A vertical line has undefined slope and equation x = constant. A horizontal line has slope zero and a vertical perpendicular. Two identical points do not specify a unique line. Inclination is reported between 0° and 180°, independent of point order.
Endpoint from a midpoint
If you know one endpoint (x₁, y₁) and the midpoint (xₘ, yₘ), the other endpoint is x₂ = 2xₘ − x₁ and y₂ = 2yₘ − y₁. For endpoint (1, 2) and midpoint (3, 6), the other endpoint is (2 × 3 − 1, 2 × 6 − 2) = (5, 10).
In every mode the steps show the slope substitution, Δy ÷ Δx reduced to lowest terms, and the intercept b = y₁ − m × x₁ with your numbers.