SOLUTION: Find the shortest distance from (2,1) to the line 3x+4y=24

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Question 905667: Find the shortest distance from (2,1) to the line 3x+4y=24
Answer by josgarithmetic(39617) About Me  (Show Source):
You can put this solution on YOUR website!
Imagine the line perpendicular to 3x+4y=24 and containing the point (2,1). The two lines intersect
at some point; and the distance between this point and (2,1) will be the distance which answers
your question.

Form the line equation 4x-3y=c.
c=4%2A2-3%2A1
c=8-3
c=5.
highlight_green%284x-3y=5%29, the perpendicular line found.
Keep the equation in this form for ease in some use of Elimination Method.

system%283x%2B4y=24%2C4x-3y=5%29
system%2812x%2B16y=96%2C12x-9y=15%29
16y-%28-9y%29=96-15
25y=81
highlight_green%28y=81%2F25%29
-
system%283x%2B4y=24%2C4x-3y=5%29
system%289x%2B12y=72%2C16x-12y=20%29
25x=92
highlight_green%28x=92%2F25%29
-
Intersection point on 3x+4y=24 is (92/25,81/25).

Distance Asked :
sqrt%28%2892%2F25-2%29%5E2%2B%2881%2F25-1%29%5E2%29
sqrt%28%2892%2F25-50%2F25%29%5E2%2B%2881%2F25-25%2F25%29%5E2%29
sqrt%28%2842%2F25%29%5E2%2B%2856%2F25%29%5E2%29
sqrt%28%281%2F25%5E2%29%2842%5E2%2B56%5E2%29%29
%281%2F25%29sqrt%2842%5E2%2B56%5E2%29
%281%2F25%29sqrt%281764%2B3136%29
%281%2F25%29sqrt%284900%29
%281%2F25%29%2A7%2A10
70%2F25=%287%2A2%2A5%29%2F%285%2A5%29
highlight%2814%2F5=2%264%2F5%29