PLEASE HELP!!!!!

1/x + 1/y = 1/18

$\frac{x+y}{xy}=\frac{1}{18}$.

$18x+18y=xy$

$xy-18x-18y=0$

Note that the expression is very close to $(x-18)(y-18)$ is if I add 324 to both sides.

$(x-18)(y-18)=324$.

324=2^2*3^4

12+18=30

27+18=45

30 and 45 give the largest result, thus x+y=30+45=75.

Since this is a symmetric equation (you can interchange x and y without changing the equation),

the minimum value will occur when x = y.

Solving 1/x + 1/y  =  1/18   --->   1/x + 1/x  =  1/18   --->   2/x  =  1/18   --->   x  =  36   --->   x + y  =  72.