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Find the sum of the \(x\) coordinates of all possible positive integer solutions to \(\frac{1}{x} +\frac{1}{y} = \frac{1}{7}\)

 Jan 30, 2019
 #1
avatar+111357 
+1

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

 

 

cool cool cool

 Jan 30, 2019
 #2
avatar+111357 
+1

Here's another way to see this :

 

1/x  +  1/y =   1/7

 

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.....

 

 

cool cool cool

 Jan 30, 2019

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