# SOLUTION: Determine the shortest distance from (9,5) to 5x + 3y =15

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 Question 258389: Determine the shortest distance from (9,5) to 5x + 3y =15Found 2 solutions by stanbon, richwmiller:Answer by stanbon(57967)   (Show Source): You can put this solution on YOUR website!Determine the shortest distance from (9,5) to 5x + 3y =15 --------------------------- The shortest distance is a perpendicular distance. --- Find the slope of the given line: y = (-5/3)x + 5 --- ANY line perpendicular to that line must have a slope = 3/5 -------------------------- Find the equation of the line with slope = 3/5 and passing thru (9,5): 5 = (3/5)9 + b 5 = (27/5) + b b = (25/5)-(27/5) b = -2/5 ======================== Equation you want: y = (3/5)x - (2/5) 5y = 3x - 2 3x - 5y = 2 ======================== Find the point of intersection of the two equations: 3x-5y = 2 5x+3y = 15 ---- Multiply 1st Eq. by 3; Multiply 2nd equation by 5: 9x - 15y = 6 25x- 15y = 75 ----------------------- Add and solve for "x": 34x = 81 x = 81/34 -- Substitute to solve for "y": 3(81/34) - 5y = 6 243/34 - 5y = 6 5y = (243-204)/34 5y = 39/34 y = (39/170) ------------- The point of intersection is (81/34 , 39/170) ------------------- Find the distance between that point and (9,5) ================================================ Cheers, Stan H. Answer by richwmiller(9144)   (Show Source): You can put this solution on YOUR website!I had to think a second about this one. They are asking for the equation that for the line which passes through (9,5) and is perpendicular to 5x + 3y =15. Then we need the intersection of the two lines and then the distance from the intersection to (9,5) so find the slope of 5x + 3y =15 3y=15-5x y=15/3-5x/3 m=-5/3 perpendicular slope would be 3/5 5=3/5(9)+b 25=27+5b -2=5b -2/5=b y=3/5x-2/5 5x + 3y =15 5y=3x-2 and 3x-2=5y 3x-5y=2 5x+3y=15 9x-15y=6 25x+15y=75 34x=81 x = 81/34, y = 35/34 (9,5) (81/34,35/34) a^2+b^2=c^2 sqrt((9-81/34)^2 +(5-35/34)^2)= 7.7 units Here are some other ways to do it. Some much shorter. http://www.worsleyschool.net/science/files/linepoint/distance.html