A Very Basic Question about a Intersecting Lines

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?
 
(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?

If a/b = a'/b' → L and L' are parallel
 
(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?
If either b or b' is 0, you can't write the expression in the form where they are in denominators. The respective line would be x=constant.
 
I wasn't paying close enough attention to what it was saying. Thanks everyone for helping though.
 
Top