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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beckden's profile pic

Posted on

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))

united07's profile pic

Posted on

but the Answer is (a-b;a+b)

 

and also you have a mistake in 1st equation. there must be - not + between a and b.

so please could you solve it once more. i will be thankfull for that..

giorgiana1976's profile pic

Posted on

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 ]).

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