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If $x^{2} + y^{2} + z^{2} = 2 (x - 2 y + 3 z) - 14$ for real $x, y, z$ , find $x + 2 y + 3 z$ .

Competition practiceAlgebra

A6
B4
C8
D2

Step-by-Step Solution

Rearranged:

(x - 1)^{2} + (y + 2)^{2} + (z - 3)^{2} = 0

, so

x = 1, y = - 2, z = 3

, and

x + 2 y + 3 z = 1 - 4 + 9 = 6

.

Answer: A) 6

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