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