Question

If 2x + 3y = 10, xy = 4, find 8x³ + 27y³.

Answer 1

Here, 2x + 3y = 10, xy = 4,

Using the identity,

a³ + b³ = (a + b)³ + 3ab(a + b)

Let’s take a = 2x and b = 3y,

Then,

(2x + 3y)³ = (2x)³ + (3y)³ + 3(2x) (3y) (2x+3y)

= 8x³ + 27y³ + 3(6xy) (2x+3y)

= 8x³ + 27y³ + 18xy(2x+3y)

Substituting the given values,

(2x + 3y)³ = (10)³

∴ (2x + 3y)³ = 1000,

18xy(2x + 3y) = 18 × 4 × 10 = 720

So,

1000 = 8x³ + 27y³ + 720

∴ 8x³ + 27y³ = 1000 − 720

∴ 8x³ + 27y³ = 280