Решите систему уравнений методом замены переменной: 3 (x - y) в квадрате + 2 (x + 2y) в квадрате = 52 (x + 2y) - x + y = 1

1. Сделаем замену:

{x - y = u;

{x + 2y = v; {3 (x - y) ^2 + 2 (x + 2y) ^2 = 5;

{2 (x + 2y) - (x - y) = 1; {3u^2 + 2v^2 = 5;

{2v - u = 1; {3u^2 + 2v^2 = 5;

{u = 2v - 1.

2. Квадратное уравнение:

3 (2v - 1) ^2 + 2v^2 = 5; 3 (4v^2 - 4v + 1) + 2v^2 = 5; 12v^2 - 12v + 3 + 2v^2 - 5 = 0; 14v^2 - 12v - 2 = 0; 7v^2 - 6v - 1 = 0; D/4 = 3^2 + 7 * 1 = 9 + 7 = 16 = 4^2; v = (3 ± 4) / 7;

1) v = (3 - 4) / 7 = - 1/7;

u = 2v - 1 = 2 * (-1/7) - 1 = - 2/7 - 1 = - 9/7;

2) v = (3 + 4) / 7 = 1.

u = 2v - 1 = 2 * 1 - 1 = 1.

3. Вернемся к исходным переменным:

{x - y = u;

{x + 2y = v; {3y = v - u;

{3x = 2u + v; {x = (2u + v) / 3;

{y = (v - u) / 3;

1) u = - 9/7; v = - 1/7;

x = (2u + v) / 3 = (-18/7 - 1/7) / 3 = - 19/21; y = (v - u) / 3 = (-1/7 + 9/7) / 3 = 8/21;

2) u = 1; v = 1;

x = (2u + v) / 3 = (2 + 1) / 3 = 1; y = (v - u) / 3 = (1 - 1) / 3 = 0.

Ответ: (-19/21; 8/21), (1; 0).