Using slopes, show that A(−2, −3), B(3, 0) and C(8, 3) are collinear. Find k if D(k, 6) also lies on this line.

1. 1 · Write what is given and what is to be found Given: A(−2, −3), B(3, 0), C(8, 3) and D(k, 6). To show: A, B, C are collinear. To find: k.
2. 2 · Draw the figure Plot A, B and C roughly. They should look as if they lie on one line.
3. 3 · Write the formula Slope m = (y₂ − y₁)/(x₂ − x₁)
4. 4 · Find slope AB and slope BC
   • Slope AB = (0 + 3)/(3 + 2) = 3/5
   • Slope BC = (3 − 0)/(8 − 3) = 3/5
5. 5 · Conclude Slope AB = slope BC and B is a common point. Therefore A, B, C are collinear.
6. 6 · Find k Slope CD = (6 − 3)/(k − 8) = 3/(k − 8). Put 3/(k − 8) = 3/5, so k − 8 = 5 and k = 13.
7. 7 · Check and box the answer A, B, C are collinear and k = 13. Check: slope AD = (6 + 3)/(13 + 2) = 9/15 = 3/5.

Scoring tip: Always write 'B is a common point'. Equal slopes alone only show that two lines are parallel.