The mathematical idea behind Slope
Slope is the vertical change divided by horizontal change between two distinct points. It is constant everywhere on a straight, nonvertical line.
Math calculator
Slope Calculator
Calculate rise over run between two coordinate points and detect vertical lines. A checkable formula accompanies slope instead of leaving an unexplained number.
Slope inputs
Known quantities
Working through Slope with numbers
From (2, 3) to (8, 15), rise is 15 − 3 = 12 and run is 8 − 2 = 6. Their ratio is 12/6 = 2.
Boundary cases 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.
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.
What a Slope result can support
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.
Reading movement in Slope
Start the Slope review with x₁. Compare x₁ with its source, then test y₁ in a second Slope run without changing the first Slope case.
Reviewing the Slope case
Before carrying Slope into another step, test Slope with a nearby round value for x₁. Retaining y₁ makes the comparison interpretable and helps separate a numerical surprise from a setup error.
Start the Slope cross-check with x₁. Apply the original y₁ and recompute Slope. Treat a different x₂ as new Slope data. That prevents its Slope from being attributed to the earlier Slope setup.
What people ask about Slope
What does a negative slope mean?
The line falls as x increases.
What is the slope of a horizontal line?
Zero, because its vertical change is zero while its run is nonzero.
Why is vertical slope undefined?
Its horizontal change is zero, which would require division by zero.
Does swapping both points change slope?
No. Both differences change sign, leaving their ratio unchanged.