If 16x² + 9y² +4z² = 24(x - y + z) - 61, then find the value of 2(xy + z) is?

4

16x² + 9y² + 4z² = 24 (x - y + z) - 61

16x² + 9y² + 4z² - 24x + 24y - 24z + 61 = 0

(4x - 3)² + (3y + 4)² + (2z - 6)² = 0

4x - 3 = 0   ⇒ x = \(\frac{3}{4}\)

3y + 4 = 0  ⇒ y = \(\frac{-4}{3}\)

2z - 6 = 0   ⇒ z = 3

Now,

2(xy + z) = 2 (\(\frac{3}{4}\) × \(\frac{-4}{3}\)) + 3) = 2( -1 + 3 ) = 4