Answer:
To find the point on the line 4x + y = 8 that is closest to the point (-3, 2), we need to first find the equation for the line perpendicular to 4x + y = 8 that passes through (-3, 2).
The slope of the line 4x + y = 8 is -4, so the slope of the line perpendicular to it is 1/4. Now we know the slope and the point (-3, 2), so we can use the point-slope form of a line to find the equation of the perpendicular line:
y - 2 = 1/4(x + 3)
y = 1/4x + 17/4
Next, we need to find the point of intersection between the line 4x + y = 8 and the perpendicular line we just found. We can solve for x and y simultaneously:
4x + y = 8
y = 1/4x + 17/4
Substituting the second equation into the first:
4x + 1/4x + 17/4 = 8
17/4x = 7/4
x = 7/17
Substituting x back into either equation, we get:
y = 1/4(7/17) + 17/4 = 39/17
So the point on the line 4x + y = 8 that is closest to the point (-3, 2) is (7/17, 39/17).