Find the sum of the \(x\) coordinates of all possible positive integer solutions to \(\frac{1}{x} +\frac{1}{y} = \frac{1}{7}\)
1/x + 1/y = 1/7
x + y = xy / 7
7 (x + y ) = xy
7 = xy / ( x + y)
x = 8 , y = 56
x = 14, y = 14 [ if x , y are not distinct ]
x = 56, y = 8
Sum = 8 + 14 + 56 = 78
Here's another way to see this :
xy = 7 (x + y)
7(x + y) - xy = 0
7x + 7y - xy = 0
7x + y(7 - x) = 0
7x = -y (7 - x)
7x = y(x - 7)
y = 7x / ( x - 7)
y = 56 when x = 8
y = 14 when x = 14
y = 8 when x = 56
etc.....
