Question

1/x -1/y = 1/60 & 1/(y+2) - 1/(x-2) = 1/60

Answer 1

1/x-1/y=1/(y+2)-1/(x-2), because they both equal 1/60, we may have the solution x=y+2 and y=x-2, which both imply y=x-2, so let's substitute for y in each equation:

1/x-1(x-2)=1/60, (x-2-x)/[x(x-2)]=1/60, -2/[x(x-2)]=1/60, -120=x(x-2), x²-2x+120=0 which has complex roots. So the assumption y=x-2 is invalid if the solution is real. 

Now let's look at this another way:

1/x-1/y=1/60, (y-x)/xy=1/60. So we need to find two whole numbers (integers) whose difference is 1 and whose product is 60. Integer pairs whose product is 60 are:

(1,60), (2,30), (3,20), (4,15), (5,12), (6,10). Their respective differences are: 59, 28, 17, 11 and 4. None meet the requirements.

But 1/60=2/120=3/180=4/240=5/300=6/360=7/420=etc. Note that 2/120 requires a difference of 2 and a product of 120. 10 and 12 fit these requirements so y=12 and x=10 giving us (y-x)/xy=2/120=1/60.

Plug these into 1/(y+2)-1/(x-2) and we get 1/14-1/8=6/112=3/56. So that doesn't work.

Try y=10 and x=12: 1/x-1/y=1/12-1/10=-1/60 and 1/(y+2)-1/(x-2)=1/12-1/10=-1/60. So that doesn't work because we get -1/60 instead of 1/60.

Now we proceed without any assumptions.

1/x-1/y=1/60=1/(y+2)-1/(x-2),

60y-60x=xy;

(x-2)(y+2)=60(x-2)-60(y+2),

xy+2x-2y-4=60x-120-60y-120,

xy=58x-58y-236.

Therefore:

60y-60x=58x-58y-236,

118x-118y=236, x-y=2, y=x-2. NOTE: This was the initial assumption, now proved to be correct.

So, 1/x-1/(x-2)=1/60,

-2/[x(x-2)]=1/60,

-120=x²-2x, x²-2x+120=0, which has no real solution. This is the same equation as we got using the initial assumption y=x-2. I suspect an error in the question:

1/x-1/y=-1/60 would give us x²-2x-120=0=(x-12)(x+10), so x=12⇒y=10; or x=-10⇒y=-12.

so, for (12,10):

1/12-1/10=-1/60 and 1/(y+2)-1/(x-2)=1/12-1/10=-1/60.

AND 

for (-10,-12):

-1/10+1/12=-1/60 and 1/(y+2)-1/(x-2)=-1/10+1/12=-1/60.

COMPLEX SOLUTION

x²-2x=-120, x²-2x+1=-119, (x-1)²=-119, x=1±√-119, y=x-2=-1±√-119, which is unlikely to be the expected solution.

REVISED QUESTION

1/x-1/y=-1/60, 1/(y+2)-1/(x-2)=-1/60. Solution is (12,10) or (-10,-12).

Alternatively: 1/y-1/x=1/60, 1/(x-2)-1/(y+2)=1/60.