Question

Determine if L1 [x,y,z]=[4,-3,2]+t[1,8,-3] and L2 [x,y,z]=[2,-19,8]+s[4,-5,-9] intersect

Answer 1

by setting the x's, the y's and the z's equal, you obtain 3 different equations in 2 unknowns.
Use the first two to find the value of s and t.
Then check if those values satisfy the third equation.