Система уравнений: (x+y) (x+y+z) = 72 (y+z) (x+y+z) = 120 (x+z) (x+y+z) = 96

1. Обозначим:

p = x + y + z;

{ (x + y) (x + y + z) = 72;

{ (y + z) (x + y + z) = 120;

{ (x + z) (x + y + z) = 96; {p (p - z) = 72;

{p (p - x) = 120;

{p (p - y) = 96.

2. Сложим все уравнения:

p (3p - x - y - z) = 72 + 120 + 96; p (3p - p) = 288; 2p^2 = 288; p^2 = 144; p = ±12.

3. Выразим x, y и z через p:

{p - z = 72/p;

{p - x = 120/p;

{p - y = 96/p; {z = p - 72/p;

{x = p - 120/p;

{y = p - 96/p.

a) p = - 12;

{z = - 12 + 72/12;

{x = - 12 + 120/12;

{y = - 12 + 96/12; {z = - 6;

{x = - 2;

{y = - 4;

b) p = 12;

{z = 12 - 72/12;

{x = 12 - 120/12;

{y = 12 - 96/12; {z = 6;

{x = 2;

{y = 4.

Ответ: (-2; - 4; - 6); (2; 4; 6).