Find the point on the line y = 5x + 4 that is closest to the origin. (x, y) = $$− 10 13​, 2 13​‍

we know that

 the point on the line y = 5x + 4 that is closest to the origin will be on the perpendicular line through the origin,

the line 
y=5x+4---------> the slope is m1=5

if two lines are perpendicular 
m1*m2=-1-----------> m2=-1/m1

so
m2=-1/5
y = (-1/5)x

so will satisfy 
(-1/5)x = 5x +4 
(-26/5)x = 4 
x = 4*(-5/26) = -10/13 
y = (-1/5)x = 2/13


The answer is
the point  (-10/13, 2/13)

see the attached figure