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

Solution: We need to minimize the function
d =

√((x - (-3))^2 + (y - 1)^2)

which represents the distance between the point (-3, 1) and any arbitrary point (x, y) on the line 6x + y = 9. To find the point on the line that is closest to (-3, 1), we need to minimize this distance function.

To begin, let's substitute y in terms of x from the equation of the line:

6x + y = 9
y = 9 - 6x

Now, substitute this expression for y in the distance function:

d = √((x - (-3))^2 + (9 - 6x - 1)^2)
d = √((x + 3)^2 + (8 - 6x)^2)

To minimize the distance function, we can take the derivative of d with respect to x and set it equal to zero:

d' = 0

To simplify the algebra, let's square both sides of the equation:

d^2 = (x + 3)^2 + (8 - 6x)^2

Now, let's take the derivative of d^2 with respect to x:

d^2' = 0
2(x + 3) + 2(8 - 6x)(-6) = 0
2x + 6 - 12(8 - 6x) = 0
2x + 6 - 96 + 72x = 0
74x - 90 = 0
74x = 90
x = 90/74
x = 45/37

Now that we have the x-coordinate, we can substitute it back into the equation of the line to find the y-coordinate:

y = 9 - 6(45/37)
y = 9 - 270/37
y = 333/37

Therefore, the point on the line 6x + y = 9 that is closest to the point (-3, 1) is approximately (45/37, 333/37).