Coordinate Geometry • Topic 3 of 4

Slope Applications

Slope measures a line’s steepness as rise over run: between two points it is (y₂ − y₁)/(x₂ − x₁). 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. Always subtract the y-values and the x-values in the same order, then reduce the fraction. Slope is the rate of change in linear models — how much y changes per unit of x — which is how the SAT often frames it in word problems. Be careful with signs when coordinates are negative, and never divide by a zero run (that signals a vertical line).

Line through (1,1) and (3,7) with rise 6 and run 2SlopeO(1, 1)(3, 7)run = 2rise = 6slope = rise/run = 6/2 = 3

✅ Solved examples

1. Find the slope through (1, 2) and (3, 8).
(8 − 2)/(3 − 1) = 6/2 = 3.
2. Find the slope through (0, 5) and (4, 1).
(1 − 5)/(4 − 0) = −4/4 = −1.
3. Find the slope through (2, 3) and (6, 3).
(3 − 3)/(6 − 2) = 0 (horizontal).
4. Find the slope through (−1, −2) and (1, 4).
(4 − (−2))/(1 − (−1)) = 6/2 = 3.

✏️ Practice — try these, take hints as needed

1. Find the slope through (1, 1) and (4, 7).
(y₂ − y₁)/(x₂ − x₁).
(7 − 1)/(4 − 1).
6/3.
2.
2. Find the slope through (0, 0) and (5, 10).
10/5.
—
—
2.
3. Find the slope through (2, 6) and (5, 0).
(0 − 6)/(5 − 2).
−6/3.
—
−2.
4. Find the slope through (3, 4) and (3, 9).
Run = 3 − 3 = 0.
Division by zero.
—
Undefined (vertical line).
5. Find the slope through (−2, 1) and (2, 9).
(9 − 1)/(2 − (−2)).
8/4.
—
2.

📝 Topic test — 8 questions

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