Question 905667
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*2-3*1}}}
{{{c=8-3}}}
{{{c=5}}}.
{{{highlight_green(4x-3y=5)}}}, the perpendicular line found.
Keep the equation in this form for ease in some use of Elimination Method.


{{{system(3x+4y=24,4x-3y=5)}}}
{{{system(12x+16y=96,12x-9y=15)}}}
{{{16y-(-9y)=96-15}}}
{{{25y=81}}}
{{{highlight_green(y=81/25)}}}
-
{{{system(3x+4y=24,4x-3y=5)}}}
{{{system(9x+12y=72,16x-12y=20)}}}
{{{25x=92}}}
{{{highlight_green(x=92/25)}}}
-
Intersection point on 3x+4y=24 is  (92/25,81/25).


Distance Asked :
{{{sqrt((92/25-2)^2+(81/25-1)^2)}}}
{{{sqrt((92/25-50/25)^2+(81/25-25/25)^2)}}}
{{{sqrt((42/25)^2+(56/25)^2)}}}
{{{sqrt((1/25^2)(42^2+56^2))}}}
{{{(1/25)sqrt(42^2+56^2)}}}
{{{(1/25)sqrt(1764+3136)}}}
{{{(1/25)sqrt(4900)}}}
{{{(1/25)*7*10}}}
{{{70/25=(7*2*5)/(5*5)}}}
{{{highlight(14/5=2&4/5)}}}