The Student
Junior Member
- Joined
- Apr 25, 2012
- Messages
- 241
(from my notes)
Consider L = {(x,y) ∈ E | ax + by + c = 0} where E is the Euclidean plane, and the two equations,
ax + by = c
a'x + b'y = c'
Assuming L (resp. L') is a proper line, a and b (resp. a', b') cannot both be zero at the same time. We all know that if L and L' are neither equal nor parallel, then L ∩ L' contains a single point. If both b, b' ≠ 0 this means that a/b ≠ a'/b', or ab' ≠ a'b.
My question: Why does b, b' ≠ 0 necessarily mean that a/b ≠ a'/b'. Couldn't b and b' still be the same number that is not 0?
Consider L = {(x,y) ∈ E | ax + by + c = 0} where E is the Euclidean plane, and the two equations,
ax + by = c
a'x + b'y = c'
Assuming L (resp. L') is a proper line, a and b (resp. a', b') cannot both be zero at the same time. We all know that if L and L' are neither equal nor parallel, then L ∩ L' contains a single point. If both b, b' ≠ 0 this means that a/b ≠ a'/b', or ab' ≠ a'b.
My question: Why does b, b' ≠ 0 necessarily mean that a/b ≠ a'/b'. Couldn't b and b' still be the same number that is not 0?