Slope Between Two Points

Slope is rise over run — how far a line climbs for each unit it travels sideways. What makes it a useful number rather than an arbitrary one is that you get the same answer whichever two points on the line you choose. That is not a coincidence; it is the defining property of a straight line, and it follows from similar triangles: any two points on the line, dropped to a horizontal and a vertical, form right triangles with the same angles, so their legs stay in the same ratio no matter how far apart you take them. A curve has no slope in this sense, because the ratio would depend on which pair you picked.

A worked instance in units a reader would hold. A wheelchair ramp is limited to a 1:12 slope, or 0.0833. A doorway 400 mm above the walk therefore needs $400/0.0833 = 4.8$ m of run, plus landings — which is why a step that looks trivial so often needs half the front yard. Drainage runs the same arithmetic in the other direction: a sanitary branch falling a quarter inch per foot has a slope of 0.0208, so 6 m of pipe drops 125 mm, and that number decides whether the fixture can reach the stack.

Slope is the ancestor of the derivative. Bring the two points closer and closer together on a curve and the ratio settles on the instantaneous rate of change at that spot, which is the whole of differential calculus in one sentence. It is also worth remembering that a slope carries units in any applied setting — dollars per kilometre, degrees per minute, metres per second. Only on a bare coordinate grid, with both axes in the same units, is it a pure number.

Three things to watch. First, keep the subtractions consistent: $(y_2 - y_1)/(x_2 - x_1)$ is fine and $(y_1 - y_2)/(x_1 - x_2)$ gives the identical answer, because negating both halves leaves the quotient alone — but mixing the two flips the sign. Second, a horizontal line has slope 0 while a vertical line has no slope at all. Its run is zero, and the quotient is undefined rather than infinite; these two get swapped constantly. Third, and most common outside the classroom: a percent grade is not an angle. Grade is $\tan\theta$ expressed as a percentage, so a 6% grade is 3.4°, not 6°, and a 100% grade is 45° rather than a cliff. Below about 10% the two are close enough that the habit forms; above it, the error grows quickly.