Verified answer

If L1 intersects L2, then their x-, y-, and z-coordinates must match. So we must have

x: 5 + 5s = 2 + 8t

y: 1 = 4 - 3t

z: 4 + 2s = 5 + 2t

or after some rearrangement,

5s - 8t = -3

3t = 3

2s - 2t = 1

The second equation immediately gives us t = 1. Putting this in the first equation and solving for s, we get s = 1. But s = 1, t = 1 does not satisfy the last equation. So L1 and L2 have no point of intersection.