Question

Solve the following pairs of equations:

`(2)/x + (3)/y = (9)/(xy)`

`(4)/x + (9)/y = (21)/(xy)`
Where x ≠ 0, y ≠ 0

Answer 1

The given equations are `(2)/x + (3)/y = (9)/(xy)` and `(4)/x + (9)/y = (21)/(xy)`

Let `(1)/x = "a" and (1)/y = "b"`
Then, we have
2a + 3b = 9ab    ....(i)
4a + 9b = 21ab  ....(ii)
Multiplying eqn. (i) by 2, we get
4a + 6b = 18ab   ....(iii)
Subtracting eqn. (iii) from eqn. (ii), we get
3b = 3ab
⇒ a = 1
⇒ `(1)/x` = 1
⇒ x = 1
Substituting the value of a in (i), we get
2(1) + 3b = 9(1)b
⇒ 2 + 3b = 9b
⇒ 6b = 2
⇒ b = `(1)/(3)`
⇒ `(1)/y = (1)/(3)`
⇒ y = 3
Thus, the solution set is (1, 3).