Find the point on the line 6x+y=9 that is closest to the point (-3,1).

 Basically, you have to find the line through (-3, 1) that is perpendicular to the line 6x + y = 9 and then find where the two lines intersect. Dropping a perpendicular from the point to the line is the shortest distance from the point to the line. 

6x + y = 9
y = -6x + 9 
slope is -6 

perpendicular would have slope 1/6
(y - 1) = (1/6) (x + 3) 
y - 1 = (1/6) x + (1/2) 
multiplying both sides by 6
6y - 6 = x + 3 
x - 6y = -9 

Where do these two lines intersect? 

From above we know given equation is y = -6x + 9
Plugging into other equation
x - 6(-6x + 9) = -9
x + 36x - 54 = -9 
37 x = 45
x = 45/37 
y = -6(45/37) + 9 
y = -270/37 + (333/37) 
y = 63/37 

point is (45/37, 63/37)