Question

Solve the following pair of equations by cross multiply method
57/x+y + 6/x-y=5
38/x+y + 21/x-y=9

Answer 1

57/(x+y) + 6/(x-y) = 5
38/(x+y) + 21/(x-y) = 9

Let's use a substitution x-y = u, x+y = v. Note that neither u nor v can be equal to zero. Then
(1) 57/v + 6/u = 5 | multiply by uv
(2) 38/v + 21/u = 6 | multiply by uv
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(1') 57u + 6v = 5uv | multiply by 9
(2') 38u + 21v = 9uv | multiply by 5
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(1") 513u + 54v = 45uv
(2") 190u + 105v = 45uv | subtract from (1")
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323u - 51v = 0
v = 513u/51 = 19u/3

Now, we get from (1'):
57u + 6*19u/3 = 5u*(19u/3)
57u + 38u = 95u^2/3
95u = 95u^2/3

As u cannot be zero, we can divide both sides by 95u:
1 = u/3
u = 3 => v = 19
So, x-y = 3, x+y = 19; these two equations have one solution: x = 11, y = 8.