Key Idea: Two lines meet where both equations are true at once. A perpendicular bisector is the line of points the same distance from A and from B, such as a road between two towns.
| Method | Best when |
|---|---|
| Substitution | One equation already gives y = ... |
| Elimination | Both equations are in ax + by = c form |
| GDC intersect | The numbers are awkward |
Perpendicular bisector of AB
- Midpoint — M = ((x₁+x₂)/2, (y₁+y₂)/2)
- Gradient of AB — m = (y₂−y₁)/(x₂−x₁)
- Perpendicular gradient — m⊥ = −1/m
- Line through M — y − yM = m⊥(x − xM)
A perpendicular bisector always passes through the midpoint, at a right angle to AB.
Company A: y = 0.30x + 25. Company B: y = 0.50x + 15. When do they cost the same? At x = 50 km, where both cost 40 dollars.
Which company is cheaper for 80 km? A, at 49 against 55. Past 50 km its lower rate per km wins.
AB has gradient 4/3. What gradient does its perpendicular bisector have? −3/4.
How do you test whether a point lies on a line? Put its coordinates into the equation and see if both sides agree.