If, 4x + 3y = 18, and 2xy = 12, and 4x > 3y, then find the value of 64x 3  - 27y 3 .

2xy = 12

So, xy = (12/2) = 6 ......(i)

4x + 3y = 18

On squaring the above equation, we get,

(4x + 3y)  ²  = 18 ²

Or, 16x ²  + 9y ²  + 2 X 4x X 3y = 324

Or, 16x ²  + 9y ²  + 24 X 6 = 324 (since, xy = 6)

Or, 16x ²  + 9y ²  = 324 - 144

So, 16x ²  + 9y ²  = 180 ....(ii)

(4x - 3y)  ²  = 16x ²  + 9y ²  - 2 X 4x X 3y

= 180 - 24 X 6 = 180 - 144 = 36 {using values from equations (i) and (ii) }

Since, 4x > 3y, so, 4x - 3y, cannot be negative.

So, 4x - 3y = 6 ....(iii)

We know that, a ³  - b ³  = (a - b) X (a ²  + b ²  + ab) .

So, (4x)  ³  - (3y)  ³  = 64x ³  - 27y ³

= (4x - 3y) X (16x ²  + 9y ²  + 4x X 3y)

Putting the value of equations (i) , (ii) and (iii) in the above equation, we get,

= 6 X (180 + 12 X 6)

= 6 X (180 + 72)

= 6 X 252 = 1,512