write two different vector equations for the plane that contains the points A(6,1,4), B(-3,7,9), C(5,0,3)

find two direction vectors

vector CA = (1,1,1)
vector BA = (9,-6,-5)

so one such equation is
r = (6,1,4) + s(1,1,1) + t(9,-6,-5)

you write a second one,
you could simply use one of the other 2 given points, or find the third direction vector.