find the shortest distance from the origin to the line y=3x-10

Question

I need the answer in exact numbers no decimals 

Answer 1

the shortest distance will be the perpendicular distance from the origin to the line y=3x-10

the equation of this line will have the negative reciprocal for its slope

y=mx+b

y=(-1/3)x+b

this line goes through the origin, (0,0)

0=(-1/3)(0)+b

0=0+b

b=0

the equation of this line is y=(-1/3)x

using this equation and y=3x-10, find the point of intersection

(-1/3)x=3x-10

3x+(1/3)x=10

(10/3)x=10

x=3, so y=3*3-10, y=9-10, y=-1

the point of intersection is (3,-1)

use the points (0,0) and (3,-1) and the distance formula to find the perpendicular distance from the origin to the point (3,-1)

√([3-0)]²+[-1-0]²)

√(3²+[-1]²)

√(9+1)

√10 is the shortest distance from the origin to the line y=3x-10

 

Answer 2

The shortest distance is along the perpendicular to y = 3x - 10

That line is y = -x/3

The point of intersection is the solution to the system

y = 3x - 10 and y = -x/3.

Determine that point.

Use Pythagoras to determine distance from (0, 0)