If x + y = 4, xy = 2, y + z = 5, yz = 3, z + x = 6 and zx = 4, then find the value of $x^3 + y^3 +z^3 - 3xy.$

153.75

If x + y = 4, xy = 2, y + z = 5, yz = 3, z + x = 6 and zx = 4

We know that,

x³ + y³ + z³ - 3xyz = (x + y + z) × (x² + y² + z² – xy – yz – zx)

(x + y)² = x² + y² + 2xy

x + y = 4, y + z = 5, z + x = 6

x + y + z = 7.5

xy + yz + zx = 2 + 3 + 4 = 9

Now,

(x + y)² = x² + y² + 2xy

x² + y²  = 16 – 4 = 12       Because..(2xy = 4)

 (y + z)² = y² + z² + 2yz

y² + z² = 25 – 6 = 19                Because...(2yz = 6)

 (x + z)² = x² + z² + 2xz

x² + z²  = 36 – 8 = 28               Because...(2zx = 8)

So,

2x² + 2y² + 2z² = 12 + 19 + 28 = 59

x² + y² + z² = 29.5

Now,

x³ + y³ + z³ - 3xyz = (x + y + z) × (x² + y² + z² – xy – yz – zx)   

= (7.5) × (29.5 – 9)

= 7.5 × 20.5

= 153.75