Final answer:
The point on the line y = 3x + 5 that is closest to the origin is (-3/2, 1/2), found by setting up the perpendicular distance from the origin to the line and solving for the intersection.
Step-by-step explanation:
To find the point on the line y = 3x + 5 that is closest to the origin, we can use the concept of perpendicular distance from a point to a line. The shortest distance between a point and a line is along the line perpendicular to it. Thus, the point on y = 3x + 5 that is closest to the origin will be at the intersection of this line and the line perpendicular to it that passes through the origin.
Since the slope of the given line is 3, the slope of the perpendicular line will be the negative reciprocal, which is -1/3. The equation of the line through the origin with this slope is y = -1/3x. To find the intersection point, we set this equal to the equation of the given line:
3x + 5 = -1/3x
3x + 1/3x = -5
10/3x = -5
x = -5 / (10/3)
x = -3/2
Plugging x back into the original line's equation:
y = 3(-3/2) + 5 = -9/2 + 5
y = -9/2 + 10/2
y = 1/2
Thus, the point on the line y = 3x + 5 closest to the origin is (-3/2, 1/2).