Coordinate Geometry • Topic 2 of 5

Slope & Intercepts

Slope measures steepness and direction: m = (y₂ − y₁)/(x₂ − x₁), often said as "rise over run". A positive slope rises left-to-right, a negative slope falls, a horizontal line has slope 0, and a vertical line has an undefined slope (the run is zero, so you would divide by zero). Keep the point order consistent — subtract the y-values and the x-values in the same order — or a sign error creeps in. The y-intercept is where the line crosses the y-axis (x = 0), and the x-intercept is where it crosses the x-axis (y = 0). In the form y = mx + b, m is the slope and b is the y-intercept, which is why that form is so useful on the GRE. Two facts unlock most line questions: parallel lines have equal slopes, and perpendicular lines have slopes whose product is −1 (they are negative reciprocals, like 2 and −1/2).

A line with a rise-over-run right triangle marking a slope of two-thirdsSlope = rise / runxy-2-1123456-2-112345Orun = 3rise = 2(1, 1)slope m = rise / run = 2 / 3

✅ Solved examples

1. Find the slope of the line through (1, 2) and (4, 11).
m = (11 − 2)/(4 − 1) = 9/3 = 3.
2. Find the slope of the line through (−2, 5) and (3, 5).
The y-values are equal, so rise = 0: m = (5 − 5)/(3 − (−2)) = 0/5 = 0. It is a horizontal line.
3. A line has slope 4. What is the slope of any line perpendicular to it?
Perpendicular slope is the negative reciprocal: −1/4 (since 4 × (−1/4) = −1).
4. Find the y-intercept of the line through (0, −3) and (2, 1).
The point (0, −3) already has x = 0, so the y-intercept is −3. (Slope = (1 − (−3))/(2 − 0) = 2, giving y = 2x − 3.)

✏️ Practice — try these, take hints as needed

1. Find the slope through (2, 3) and (6, 15).
Rise over run.
(15 − 3)/(6 − 2).
12/4.
3
2. Find the slope through (−1, 4) and (2, −5).
Keep order consistent.
(−5 − 4)/(2 − (−1)).
−9/3.
−3
3. What is the slope of a vertical line through (3, 1) and (3, 8)?
Run = 3 − 3 = 0.
Dividing by zero.
Slope is not defined.
Undefined
4. A line has slope −2/3. What is the slope of a line parallel to it?
Parallel means equal slope.
No change.
Same value.
−2/3
5. A line has slope 5. Find the slope of a perpendicular line.
Negative reciprocal.
Flip and change sign.
Product must be −1.
−1/5

📝 Topic test — 8 questions

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