Let a point on 6x+y = 9 be (a,b)
The line formed by (a,b) and (-3,1) will be purpendicular to 6x+y = 9 when (a,b) is the closest point.
Equation of the line between (a,b), (-3,1)
(y-b)/(x-a) = (y-1)/(x+3)
y(3+a) = x(b-1)+3b+a
Since the lines are perpendicular the multiplication of thier gradients will equal -1.
6(b-1) = (3+a)
6b-a = 9------(1)
Also (a,b) is on the line 6x+y = 9
6a+b = 9-----(2)
solving (1) and (2) will give;
a=45/37 and b= 63/37
So the closest point is (45/37,63/37)