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Question
Solve the following pairs of equations:
`(2)/(x + 1) - (1)/(y - 1) = (1)/(2)`
`(1)/(x + 1) + (2)/(y - 1) = (5)/(2)`
Solution
The given equations are `(2)/(x + 1) - (1)/(y - 1) = (1)/(2)` and `(1)/(x + 1) + (2)/(y - 1) = (5)/(2)`.
Let `(1)/(x + 1) = "a" and (1)/(y - 1) = "b"`
Then, we have
2a - b = `(1)/(2)` ....(i)
a + 2b = `(5)/(2)` ....(ii)
Multiplying eqn. (i) by 2, we get
4a - 2b = 1 ....(iii)
Adding eqns. (ii) and (iii), we get
5a = `(7)/(2)`
⇒ a = `(7)/(10)`
⇒ `(1)/(x + 1) = (7)/(10)`
⇒ 10 = 7x + 7
⇒ 7x = 3
⇒ x = `(3)/(7)`
Substituting the value of a in eqn. (iii), we get
`4 xx (7)/(10) - 2"b"` = 1
⇒ `(14)/(5) - 2"b"` = 1
⇒ 2b = `(14)/(5) - 1 = (9)/(5)`
⇒ b = `(9)/(10)`
⇒ `(1)/(y - 1) = (9)/(10)`
⇒ 10 = 9y - 9
⇒ 9y = 19
⇒ y = `(19)/(9)`
Thus, the solution set is `(9/10, 19/9)`.
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