Find the point on the line 5x+7y+1=0 which is closest to the point (−1,−3).

The perpendicular line through (-1, -3) can be written as
  7(x+1) -5(y+3) = 0
The intersection of this line with the original is
  (x, y) = (51/74, -47/74)

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In standard form, the two lines can be written as
  5x +7y = -1
  7x -5y = 8
By Cramer's rule,
  x = (7·8 - (-1)(-5))/(7·7 -(-5)·5) = 51/74
  y = (-1·7 -8·5)/74 = -47/74