This was one of the questions on my midterm that I was unable to answer and I still don't know how this is done. The question gives us a plane in vector form and 2 lines also in vector form. Our task was to come up with a system of linear equations to determine whether or not the intersection of these three sets is non-empty.
The answer is in the image, I just want to know how this was done. I would be very grateful for an explanation.
Set A refers to a plane while Sets B and C are lines.
In the equation for plane A mark parameters t and s as $ x_1 $ and $ x_2 $. In the equation for line B mark parameter t as $ x_3 $. In the equation for line C mark parameter t as $ x_4 $.
Each set of three equations corresponds to one of the following
A - B = 0
B - C = 0
A - C = 0.