SOLUTION: Find the distance from point A(−14, 5) to the line −x+2y = 14. Round your answer to the nearest tenth.

Algebra ->  Linear-equations -> SOLUTION: Find the distance from point A(−14, 5) to the line −x+2y = 14. Round your answer to the nearest tenth.      Log On


   

Question 1100406: Find the distance from point A(−14, 5) to the line −x+2y = 14. Round your answer to the nearest tenth.
Answer by ikleyn(52781) About Me  (Show Source):
You can put this solution on YOUR website!
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There is a beatiful formula to find the distance from the given point to the given straight line in a coordinate plane.

This formula is: 

    Let the straight line in a coordinate plane is defined in terms of its linear equation 


        a*x + b*y + c = 0,


    where  a,  b and  c  are real numbers,  and let  P = P(x%5B0%5D,y%5B0%5D)  is the point in the coordinate plane 
    with the coordinates  x%5B0%5D,  y%5B0%5D.  Then the distance from the point  P  to the straight line is equal to


        d = abs%28a%2Ax%5B0%5D+%2B+b%2Ay%5B0%5D+%2B+c%29%2Fsqrt%28a%5E2+%2B+b%5E2%29.


See the lesson  The distance from a point to a straight line in a coordinate plane  in this site.


So, in your case  a= -1,  b= 2,  c= -14   (after moving  "c"  from the righ side to the left),  x%5B0%5D= -14,  y%5B0%5D= 5.

Substituting to the formula gives you the distance

d = abs%28%28-1%29%2A%28-14%29+%2B+2%2A5+-14%29%2Fsqrt%28%28-1%29%5E2%2B2%5E2%29 = abs%2814%2B10-14%29%2Fsqrt%281+%2B+4%29 = 10%2Fsqrt%285%29 = %2810%2Asqrt%285%29%29%2F5%29 = 2%2Asqrt%285%29 = 4.5 (approximately; rounded to the nearest tenth).


Solved.


Again,  for the proof of the formula and other details look into the above referred lesson.