Question

Solve for x and y:
`5/(x+1) + 2/(y−1) = 1/2, 10/(x+1) - 2/(y−1) = 5/2,  where  x ≠ 1, y ≠ 1.`

Answer 1

The given equations are
`5/(x+1) + 2/(y−1) = 1/2`   ……(i)
`10/(x+1) - 2/(y−1) = 5/2`   ……(ii)
Substituting `1/(x+1) = u and 1/(y−1)` = v, we get:
`5u - 2v = 1/2`   ……..(iii)
`10u + 2v = 5/2`    …….(iv)
On adding (iii) and (iv), we get:
15u = 3
`⇒u = 3/15 = 1/5`
`⇒ 1/(x+1)  = 1/5 ⇒ x + 1 = 5 ⇒ x = 4`
On substituting u = `1/5` in (iii), we get
`5 × 1/5 - 2v = 1/2 ⇒ 1 – 2v = 1/2`
`⇒2v = 1/2 ⇒v = 1/4`
`⇒ 1/(y−1) = 1/4 ⇒ y – 1 = 4 ⇒ y = 5`
Hence, the required solution is x = 4 and y = 5.