Solve simultaneous linear equations: (i) (8/27)x3 – (1/8)y3

Solve simultaneous linear equations:

(i) (8/27)x³ – (1/8)y³
(ii) x⁶ + 63x³ – 64

Solution:

(i) (8/27)x³ – (1/8)y³
Above terms can be written as,
((2/3)x)³ – (½y)³
We know that, a³ – b³ = (a – b) (a² + ab + b²)
((2/3)x – ½y) [(2/3)x + (2/3)x (1/2)y + ((1/2)y)²]
((2/3)x – (1/2)y) [(4/9)x² + (xy/3) + (y²/4)]
(ii) x⁶ + 63x³ – 64
x⁶ + 63x³ – 64
Above terms can be written as,
x⁶ + 64x³ – x³ – 64
Take out common in all terms,
x³ (x³ + 64) – 1(x³ + 64)
(x³ + 64) (x³ – 1)
(x³ + 4³) (x³ – 1³)
We know that, a³ – b³ = (a – b) (a² + ab + b²) and a³ + b³ = (a + b) (a² – ab + b²)
So, (x + 4) [x² – 4x + 4²] (x – 1) [x² + x + 1²]
(x + 4) (x² – 4x + 16) (x – 1) (x² + x + 1)

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