3x + 4y -z = 10 is the equation for a plane with normal vecor n = 3,4,-1.

A parallel plane P(x,y,z) must have the same normal vector (any scalar multiple but 0). If it is supposed to go through the origin, then P(0,0,0)=0.

If the equation for a plane is Ax +By +Cz + D = 0, then it is easy to find D so it passes through the origin.

A vector parallel to the line L is v1=(dx/dt, dy/dt, dz,dt). You have a another point P, which is not located on the line L,

so you can find a vector between any point on the line L and P, lets call it v2.

Now you have 2 vectors, v1 and v2, they lie in the plane containing P and L, so if you can find a vector perpendicular to both of them,

then you have a normal vector for the plane. Then it is easy to find D by substituting e.g. P in the resulting equation.

If they intersect then there is a point K =(7-4b, 3+6b, 9+5b), that satisfies the equation for the plane P(x,y,z) = 17 = P(K).

Hope that helps