Overview
Slope measures vertical change per horizontal change. Use the same point order in numerator and denominator so signs agree. Horizontal lines have zero slope; vertical lines have undefined slope because horizontal change is zero.
Consider this example: What is the slope of the line through (1, 2) and (5, 10)? The rise is 8 and the run is 4, so slope = 8/4 = 2. The reciprocal 1/2 reverses rise and run.
The method also handles a different case: What is the slope through (−3, 4) and (2, −6)? The vertical change is −10 and horizontal change is 5. The slope is −2; the line falls as x increases. A positive 2 usually comes from reversing just one subtraction.
For a further application, consider: The slope through (2, 3) and (6, k) is −3/2. What is k? The run is 4, so the rise must be (−3/2)(4) = −6. Thus k − 3 = −6 and k = −3. The rise −6 is not the final y coordinate.
Reversing only one subtraction changes a slope’s sign. Treating a vertical line as slope zero confuses no horizontal change with no vertical change. For a new pair of points, sketch their relative positions and predict whether the line rises or falls to the right. Check the quotient’s sign against that sketch.
A coordinate plot gives a second way to check the same rise and run. In the plotted example through (1, 2) and (5, 10), horizontal ticks count by one and the labeled vertical ticks count by two. Moving from the first point to the second changes x by 5 − 1 = 4 and y by 10 − 2 = 8. Thus the slope is 8/4 = 2. Four vertical grid spaces here represent eight coordinate units; counting spaces without reading the labels would halve the rise. Reversing the order of both point differences gives (2 − 10)/(1 − 5) = (−8)/(−4) = 2 as a self-check. Compare coordinates, not the angle of a line on an arbitrary display.