Question

Solve for x and y:
4x + 6y = 3xy, 8x + 9y = 5xy

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 = 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) = 2`
`⇒ 2x = 6 ⇒ x = 6/2 = 3`
Hence, the required solution is x = 3 and y = 4.