Slope from Two Points
Given any two points on a line, one formula gives you its steepness: subtract the y’s, subtract the x’s, divide — and always simplify. This is the prerequisite for every line equation you’ll ever write.
1 The slope formula
Slope measures steepness as rise over run: how far y climbs for each step in x. Two points give you both numbers at once.
Subtract in the same order top and bottom — y’s in the same order as the x’s. A positive slope climbs left to right; a negative slope falls.
2 Worked examples
Four computations, from the straightforward to the two edge cases every student must know.
Find the slope through (1, 2) and (5, 10).
m = (10 − 2)/(5 − 1) = 8/4 = 2
The line climbs 2 units of y for every 1 unit of x.
Find the slope through (−2, 5) and (4, −7).
m = (−7 − 5)/(4 − (−2)) = (−12)/6 = −2
Parenthesize the negatives first; the formula does the rest.
Find the slope through (3, 4) and (−5, 4).
m = (4 − 4)/(−5 − 3) = 0/(−8) = 0
No rise at all: the line is flat. Zero on top means slope zero.
Find the slope through (2, 1) and (2, 9).
m = (9 − 1)/(2 − 2) = 8/0 — division by zero is undefined.
A vertical line is infinitely steep: it has no slope. Never write “slope = 0” here.
3 Common mistakes
Three traps, each with the wrong version and the fix.
Wrong: m = (x2 − x1)/(y2 − y1) = run/rise. Right: slope is rise over run — y’s on top, x’s on the bottom.
Wrong: (y2 − y1)/(x1 − x2) — the bottom undoes the top’s order. Right: pick an order (say, “second minus first”) and use it for both the y’s and the x’s.
Wrong: “the line through (2, 1) and (2, 9) has slope 0.” Right: 0 on top → slope 0 (flat). 0 on the bottom → undefined (vertical). The zero’s address decides.
4 Quick checks
Try these yourself, then reveal the answer.
5 Key points
Remember
- m = (y2 − y1)/(x2 − x1) — same subtraction order top and bottom.
- Always simplify the fraction: −6/4 is −3/2.
- 0 on top → slope 0 (horizontal). 0 on the bottom → undefined (vertical).
- Sign tells direction: positive climbs, negative falls.
- Slope is a rate: “3 feet up for every 36 feet across” is the same idea as rise over run.