Before going into the details you may want to see my web site

about intersecting and skew lines in 3 space --notes and animations.

Vector Valued Functions

Now with your problem --you can think of the lines as the trajectories of particles. They may either collide in which case they reach the pt of int at the same time. The paths could cross but the particles reach the pt of int at different times whichj is why you need 2 variables.

(-1/2 t, 2/3 t +1, t- 3/2)

(s- 3/2, 2s+3, 3s+ 9/2)

equating the x coord -1/2t = s- 3/2

t= -2s+3

equate the y cooords

2/3t +1 = 2s +3

2/3(-2s+3)+1 = 2s + 3

-4s + 9 = 6s +9

s= 0

so on L2

(s- 3/2, 2s+3, 3s+ 9/2) -> (-3/2,3,9/2)

since t= -2s+3 it follows when s = 0 t= 3

On L1

(-1/2 t, 2/3 t +1, t- 3/2) -> (-3/2,3,9/2) as it should