Calculate rise over run between two coordinate points and detect vertical lines. A checkable formula accompanies slope instead of leaving an unexplained number.
From (2, 3) to (8, 15), rise is 15 − 3 = 12 and run is 8 − 2 = 6. Their ratio is 12/6 = 2.
Slope is the vertical change divided by horizontal change between two distinct points. It is constant everywhere on a straight, nonvertical line.
The calculation treats x₁, y₁, x₂ and y₂ according to their labels, not as interchangeable values in slope.
Choose an order for the points, subtract the first y from the second y, subtract the first x from the second x, and divide. Reduce a fractional slope when an exact ratio is useful. A hand-worked extension of Slope is their midpoint.
Subtract coordinates in the same direction in numerator and denominator. Reversing only one subtraction flips the sign incorrectly. A zero run describes a vertical line and makes numerical slope undefined. If the Slope assumptions do not fit, consider distance between the points.
Save extra digits internally if slope will become an input to another calculation.
Rates of change appear in graphs, elevation profiles, unit costs, motion, and trend lines. A slope of 2 means y rises by 2 units whenever x increases by 1 unit. A related application of Slope is solve a linear equation.
A field name is part of the formula. Match the problem's quantities to x₁, y₁, x₂ and y₂, then check that they share the scale assumed by the slope relationship.
Translating both points by the same horizontal and vertical amounts leaves slope unchanged. As the run approaches zero with nonzero rise, the slope's magnitude grows without reaching a finite vertical-line value. This is a stronger check than judging the answer only by how many decimal places it shows.
Match the reported precision to x₁, y₁, x₂ and y₂, not to the number of digits the browser can display. Preserve an exact form when it communicates the slope structure more clearly than a decimal.
Slope measures direction and rate, not segment length. Two segments can share a slope while having very different distances between their endpoints. The formula panel makes the chosen definition explicit.
The minimum audit trail is short: x₁, y₁, x₂ and y₂, their units, and the formula beside the answer. It is enough to distinguish this calculation from a similar-looking shortcut.
The line falls as x increases.
Zero, because its vertical change is zero while its run is nonzero.
Its horizontal change is zero, which would require division by zero.
No. Both differences change sign, leaving their ratio unchanged.