Find the distance for (3,5) and y=3x+2

There are standard formulas for this type of problem. However, it can also be solved by a combination of simple steps.

First, the shortest distance from the point (3,5) to the line y = x +4 (line1) will be along a straight line perpendicular to line1. Give the perpendicular line the name line2. Since the slope of line1 is 1, the slope of line2 will be -1.

Second, since line2 must go through (3,5) and also have a slope of -1, the point slope form can be used for line2:
y - 5 = (-1) (x -3)
So the equation of line2 is y = -x +8.

Third, the point of intersection of line1 and line2 can be found by solving the set of equations:
y = x +4
y = -x + 8
The solution of this set of two equations is x = 2, y = 6 i.e. the point (2,6) .

Fourth, the distance formula can be used to find the distance between (3,5) and (2,6)
d = sqrt( (3-2)2 + (5-6)2 ) = sqrt(2)
This is the desired distance.