Parametric equation of intersection of two lines?

Question

Mathematics

We need to check the intersection of two lines in 3D. I am 0 in mathematics. I found these equations:
Ax + (Bx - Ax) * t1 = C.x + (Dx - C.x) * t2
Ay + (By - Ay) * t1 = C.y + (Dy - С.y) * t2
Az + (Bz - Az) * t1 = С.z + (Dz - С.z) * t2 It is
necessary to check:
If in at least one case it turns out that such t1 and t2 do not exist, then do not intersect.
Otherwise, if in at least one case an answer is obtained for t, we check whether it fits in the remaining of the three equations. If yes - found, if not - do not intersect.
If neither of the two conditions is met, then the lines coincide.
So I want to know what t1 and t2 are? What are they needed for? And how to "check" whether they exist or not?
You can only, please, describe exactly this case here (if it’s really detailed, then with an example in C ++), and not start being smart, throwing links to books, etc.