L1: [x,y,z] = [1,-2,4,] + s[1,1-3]

L2: [x,y,z] = [4,-2,k] + t[2,3,1]

Determine a value of k that is 10 units away from the plane with the equation 10x-7y+Z-28=0

Your question doesn't make a lot of sense to me.

"Determine a value of k that is 10 units away from the plane ..." ??
How can a number be 10 units away from a plane?
A point can be, or a line could be if that line would be parallel to the plane.

Let's check if L2 is parallel to the plane
direction of line = (2,3,1)
normal to the plane = (10,-7,1)
dot product = 20 - 21 + 1 = 0
Well, what do you know, the line is parallel.
In that case , look at the solution to the question found under "vectors" in the Related Questions below.

btw, what is the purpose of giving us L1 ?