Question

Solve for x and y:
`10/(x+y) + 2/(x−y) = 4, 15/(x+y) - 9/(x−y) = -2, where  x ≠ y, x ≠ -y.`

Answer 1

The given equations are
`10/(x+y) + 2/(x−y) = 4 `   ……(i)
`15/(x+y) - 9/(x−y) = -2`   ……(ii)
Substituting `1/(x+y) = u and 1/(x−y)` = v in (i) and (ii), we get:
10u + 2v = 4 ……..(iii)
15u - 9v = -2 …….(iv)
Multiplying (iii) by 9 and (iv) by 2 and adding, we get:
90u + 30u = 36 –4

⇒120u = 32
`⇒u = 32/120 = 4/15`
`⇒x + y = 15/4   (∵1/(x+y)=u)`  …..(v)
On substituting u = `4/15` in (iii), we get:
`10 × 4/15 + 2v = 4`
`8/3 + 2v = 4`
⇒2v = 4 - 83 = 43
`⇒v = 2/3`
`⇒ x – y = 3/2       (∵1/(x−y) =v) `     …..(vi)
Adding (v) and (vi), we get
`2x = 15/4 + 3/2 ⇒2x = 21/4 ⇒x = 21/8`
Substituting x`  = 21/8 `in (v), we have
`21/8 + y = 15/4 ⇒y = 15/4 - 21/8 = 9/8`
Hence, x =` 21/8 and y =9/8.`