SOLUTION: find the distance from (-6,-3) to the line defined by y=3x+5 .

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Question 749603: find the distance from (-6,-3) to the line defined by y=3x+5 .
Answer by josgarithmetic(39623) About Me  (Show Source):
You can put this solution on YOUR website!
A line containing (-6,-3) and is perpendicular to y=3x+5 also contains the point on both lines; this is the point closest to (-6,-3) on y=3x+5.

You want the line y=-%281%2F3%29x%2Bb which contains (-6,-3).
b=y%2B%281%2F3%29x
b=-3+(1/3)(-6)
b=-3-2
b=-5.
The perpendicular line is y=-%281%2F3%29x-5

What is the intersection of y=-%281%2F3%29x-5 with y=3x%2B5 ?
-%281%2F3%29x-5=3x%2B5
-3x-%281%2F3%29x=10
3x%2B%281%2F3%29x=-10
9x%2B1x=-30
10x=-30
x=-3
Substituting this y=3%28x%29%2B5
y=3%28-3%29%2B5
y=-9%2B5
y=-4.

That is the point (-3,-4). This is the point on BOTH lines, shortest distance on the y=3x+5 to (-6,-3).
How far is (-3,-4) from (-6,-3)?

Distance: sqrt%28%28-3-%28-6%29%29%5E2%2B%28-4-%28-3%29%29%5E2%29
sqrt%283%5E2%2B1%5E2%29
highlight%28sqrt%2810%29%29------------this is how far is y=3x+5 to (-6,-3).