Find the value of X and Y. (a/(x+y))-(b/(x-y))=1 (b/(x+y))+(a/(x-y))=((a²-b²)/2ab)

To add or subtract two fractions, they must have the same
denominator.

For the first and for the 2nd equations, the
common denominator is (x-y)(x+y). This product returns the difference of squares x^2 -
y^2.

The 1st equation will
become:

a(x-y) + b(x+y) = x^2 -
y^2

The 2nd equation will
become:

b(x-y) + a(x+y) = (a^2 - b^2)(x^2 - y^2)/2ab
=> x^2 - y^2 = 2ab*[b(x-y) + a(x+y)]/(a^2 -
b^2)

(a^2 - b^2)*[a(x-y) + b(x+y)] = 2ab*[b(x-y) +
a(x+y)]

a^3*(x-y) + a^2*b(x+y) - ab^2*(x-y) - b^3*(x+y) =
2ab^2(x-y) + 2a^2*b(x+y)

-3ab^2*(x-y) + a^3*(x-y) =
a^2*b(x+y) + b^3*(x+y)

(x-y)(a^3 - 3ab^2) = (x+y)(b^3+
a^2b)

a^3*x - 3ab^2*x - a^3*y + 3ab^2*y = b^3*x+ a^2b*x +
b^3*y+ a^2b*y

a^3*x - 3ab^2*x - b^3*x- a^2b*x = b^3*y+
a^2b*y + a^3*y - 3ab^2*y

x(a^3 - a^2b - 3ab^2 - b^3) =
y(b^3 + a^2b - 3ab^2 + a^3)

x = y(b^3 + a^2b - 3ab^2 +
a^3)/(a^3 - a^2b - 3ab^2 - b^3)

Therefore, the value of x
will be substituted in the 1st equation:

ay[(b^3 + a^2b -
3ab^2 + a^3)/(a^3 - a^2b - 3ab^2 - b^3)-1] + by[(b^3 + a^2b - 3ab^2 + a^3)/(a^3 - a^2b -
3ab^2 - b^3)+1] = y^2[(b^3 + a^2b - 3ab^2 + a^3)/(a^3 - a^2b - 3ab^2 - b^3) - 1][(b^3 +
a^2b - 3ab^2 + a^3)/(a^3 - a^2b - 3ab^2 - b^3) + 1]

y[a(b^3
+ a^2b - 3ab^2 + a^3)/(a^3 - a^2b - 3ab^2 - b^3)-a + b(b^3 + a^2b - 3ab^2 + a^3)/(a^3 -
a^2b - 3ab^2 - b^3)+b ] = y^2[(b^3 + a^2b - 3ab^2 + a^3)/(a^3 - a^2b - 3ab^2 - b^3) -
1][(b^3 + a^2b - 3ab^2 + a^3)/(a^3 - a^2b - 3ab^2 - b^3) +
1]

y^2[(b^3 + a^2b - 3ab^2 + a^3)/(a^3 - a^2b - 3ab^2 -
b^3) - 1][(b^3 + a^2b - 3ab^2 + a^3)/(a^3 - a^2b - 3ab^2 - b^3) + 1] - y[a(b^3 + a^2b -
3ab^2 + a^3)/(a^3 - a^2b - 3ab^2 - b^3)-a + b(b^3 + a^2b - 3ab^2 + a^3)/(a^3 - a^2b -
3ab^2 - b^3)+b ] = 0

y1 =
0

y[(b^3 + a^2b - 3ab^2 + a^3)/(a^3 - a^2b - 3ab^2 - b^3) -
1][(b^3 + a^2b - 3ab^2 + a^3)/(a^3 - a^2b - 3ab^2 - b^3) + 1] = [a(b^3 + a^2b - 3ab^2 +
a^3)/(a^3 - a^2b - 3ab^2 - b^3)-a + b(b^3 + a^2b - 3ab^2 + a^3)/(a^3 - a^2b - 3ab^2 -
b^3)+b ]

y2 = [a(b^3 + a^2b - 3ab^2 + a^3)/(a^3 - a^2b -
3ab^2 - b^3)-a + b(b^3 + a^2b - 3ab^2 + a^3)/(a^3 - a^2b - 3ab^2 - b^3)+b ]/[a(b^3 +
a^2b - 3ab^2 + a^3)/(a^3 - a^2b - 3ab^2 - b^3)-a + b(b^3 + a^2b - 3ab^2 + a^3)/(a^3 -
a^2b - 3ab^2 - b^3)+b ]

x =
0

x = y2*(b^3 + a^2b - 3ab^2 + a^3)/(a^3 - a^2b - 3ab^2 -
b^3)

The values of x and y are: (0;0) or
(y2*(b^3 + a^2b - 3ab^2 + a^3)/(a^3 - a^2b - 3ab^2 - b^3) ; [a(b^3 + a^2b - 3ab^2 +
a^3)/(a^3 - a^2b - 3ab^2 - b^3)-a + b(b^3 + a^2b - 3ab^2 + a^3)/(a^3 - a^2b - 3ab^2 -
b^3)+b ]/[a(b^3 + a^2b - 3ab^2 + a^3)/(a^3 - a^2b - 3ab^2 - b^3)-a + b(b^3 + a^2b -
3ab^2 + a^3)/(a^3 - a^2b - 3ab^2 - b^3)+b ]).