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`

⇒ `u = 4/15`

⇒ `x + y = 15/4`   `(∵ 1/(x + y) = u)`   ...(v)

On substituting `u = 4/15` in (iii), we get:

`10 xx 4/15 + 2v = 4`

`8/3 + 2v = 4`

⇒ `2v = 4 - 8/3 = 4/3`

⇒ `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`

⇒ `y = 9/8`

Hence, `x = 21/8` and `y = 9/8`.