we know that
the point p on the line y=5x that is closest to the point (78 0) will be on the line perpendicular to the line y=5x and go through the point (78 0)
step 1
find the slope m perpendicular to the line y=5x
m1=5
if two lines are perpendicular
m1*m2=-1
m2=-1/5
step 2
with m=-1/5 and the point (78,0) find the equation of a line
y-y1=m*(x-x1)-----> y-0=(-1/5)*(x-78)----> y=-0.20x+15.6
step 3
resolve the equation systems
y=5x
y=-0.20x+15.6
using a graph tool
see the attached figure
the solution is the point
x=3
y=15
the answer is
the point (3,15)