Question

Solve for x and y:

4x + 6y = 3xy, 8x + 9y = 5xy (x ≠ 0, y ≠ 0)

Answer 1

The given equations are:

4x + 6y = 3xy  ...(i)

8x + 9y = 5xy   ...(ii)

From equation (i), we have:

`(4x + 6y)/(xy) = 3`

⇒ `4/y + 6/x = 3`   ...(iii)

For equation (ii), we have:

`(8x + 9y)/(xy) = 5`

⇒ `8/y + 9/x = 5`   ...(iv)

On substituting `1/y = v` and `1/x = u`, we get:

4v + 6u = 3   ...(v)

8v + 9u = 5  ...(vi)

On multiplying (v) by 9 and (vi) by 6, we get:

36v + 54u = 27   ...(vii)

48v + 54u = 30   ...(viii)

On subtracting (vii) from (viii), we get:

12v = 3 

⇒ `v = 3/12`

⇒ `v = 1/4`

⇒ `1/y = 1/4`

⇒ y = 4

On substituting y = 4 in (iii), we get:

`4/4 + 6/x = 3`

⇒ `1 + 6/x = 3`

⇒ `6/x = (3 - 1)`

⇒ `6/x = 2`

⇒ 2x = 6 

⇒ `x = 6/2`

⇒ x = 3

Hence, the required solution is x = 3 and y = 4.