Given positive integers $x$ and $y$ such that $\frac{1}{x} + \frac{1}{2y} = \frac{1}{7}$, what is the least possible value of $xy$?
+128470 \(\frac{1}{x} + \frac{1}{2y} = \frac{1}{7}\)
[ x + 2y ] 1
_______ = ___
2xy 7
7 [ x + 2y ] = 2xy
7x + 14y = 2xy
14y - 2xy = - 7x
2xy - 14y = 7x
y [2x - 14 ] = 7x
y = 7x
_______
2x -14
If
x = 8 y = 28
x = 14 y = 7
x = 56 y = 4
We can stop here since if y = 3 then
3(2x - 14 = 7x
6x - 42 = 7x
-42 = x
And x will also be negative if y = 2 or y = 1
So.....the minimum value of xy = 14 * 7 = 98
