Question

what are 2 sets are not parallel lines

Answer 1

I’m not completely sure what you mean by “2 sets are not parallel lines,” so I’ll cover the two most common interpretations.

1) Two distinct lines in the plane that are not parallel (they intersect)
- Example (in slope-intercept form):
- L1: y = 2x + 1
- L2: y = −x + 3
- Why not parallel: their slopes are different (2 vs −1), so they meet at one point.
- Intersection point: solve 2x + 1 = −x + 3 → 3x = 2 → x = 2/3; then y = 2(2/3)+1 = 7/3. So they intersect at (2/3, 7/3).

2) Two lines in three-dimensional space that are not parallel and also do not intersect (skew lines)
- Example (parametric form):
- L1: (x,y,z) = (0,0,0) + t(1,0,0) = (t,0,0)
- L2: (x,y,z) = (0,1,1) + s(0,1,0) = (0, 1+s, 1)
- Why not parallel: direction vectors (1,0,0) and (0,1,0) are not scalar multiples, so lines are not parallel.
- Why they don’t intersect: there is no t,s making (t,0,0) = (0,1+s,1) because z-coordinates 0 ≠ 1. Such lines are called skew.

If you meant something else (e.g., sets of equations, set notation, or a specific problem), tell me and I’ll tailor the answer.