You can equate x's, y's and z's and get three equations and three variables. Note that the value of t of the two lines don't have to be the same at the intersection point.

The vector (2, 1, 4) lies in the first line, and (a, -2, 1) lies in the second line. Once you know the intersection point and two vectors in the plane, you can use these methods to construct the plane equation.