When I try to solve this system, the first thing I did was look at the first equation. I isolated D, D = -4A + 4B, and then substituted that back into all of the other equations. When I did this, all of my new equations were multiples of each other, which means that they were all coincident, so there would be infinite solutions.
When I solved the system I got the solution
these are true for all values of s and t.
This would generate an entire family of planes that would intersect in the line given. Think about why there are an infinite number of planes.
If you want you can check a few. example let s=1 and t=1
we get A=1, B=1 C=1, D=0 so for all k.