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

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a/(x+y) - b/(x-y) = 1
b/(x+y) + a/(x-y) = (a^2-b^2)/(2ab)

Let u = (x+y) and v = (x-y) and c = (a^2+b^2)/(2ab)

(1) a/u - b/v = 1
(2) b/u +a/v = c so  multiply (1) by a and (2) by b

(3) a^2/u - ab/v = a
(4) b^2/u +ab/v = bc   Add together to get

(a^2+b^2)/u =  bc+a or

u = (a^2 + b^2)/(bc + a)

Substituting into (1) we get

a/((a^2+b^2)/(bc+a)) - b/v = 1
a(bc+a)/(a^2+b^2) - 1 = b/v
(abc +a^2 - a^2 + b^2)/(a^2+b^2) = b/v
(abc + b^2)/(a^2+b^2) = b/v
v = (a^2+b^2)/(ac + b)

u+v = 2x and u-v=2y so

2x = (a^2 + b^2)/(bc + a) + (a^2+b^2)/(ac+b)
x = 1/2(a^2+b^2)(1/(bc+a)+1/(ac+b))
x = 1/2(a^2+b^2)(((ac+b)+(bc+a))/((ac+b)(bc+a)))
x = 1/2(a^2+b^2)(4ab(a+b)(c+1))/(4ab(ac+b)(bc+a))
x = (a^2+b^2)(a+b)(2abc + 2ab)/((2abc+2b^2)(2abc+2a^2))

Now 2abc = a^2+b^2 so
x = (a^2+b^2)(a+b)(a^2 + 2ab + b^2)/((a^2 + 3b^2)(3a^2 + b^2))
x = (a^2+b^2)(a+b)^3/((a^2+3b^2)(3a^2+b^2))

2y = (a^2 + b^2)/(bc + a) - (a^2 + b^2)/(ac+b)
y = 1/2 (a^2+b^2)(1/(bc+a) - 1/(ac+b))
y = 1/2(a^2+b^2)(((ac+b)-(bc+a))/((ac+b)(bc+a)))
y = 1/2(a^2+b^2)(4ab(a-b)(c-1))/(4ab(ac+b)(bc+a))
y = (a^2+b^2)(a-b)(2abc - 2ab)/((2abc+2b^2)(2abc+2a^2))

Now 2abc = a^2+b^2 so
y = (a^2+b^2)(a-b)(a^2 - 2ab + b^2)/((a^2 + 3b^2)(3a^2 + b^2))
y = (a^2+b^2)(a-b)^3/((a^2+3b^2)(3a^2+b^2))

So the answer is

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

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