Вправи: 1137

1137. Розв’яжи у цілих додатних числах систему.

а) x + y + z = 12; 2x + y + 12z = 12; z = 12 – x – y; 2x + y + 12 (12 – x – y) = 12; 2x + y + 6 – 12x – 12y = 12; 32х + 12у = 12 – 6; 32х + 12у = 6|• 2;3х + y = 12;y = 12 – 3х; x + 12 – 3х + z = 12; –2x + z = 0; z = 2x; x = 1; y = 12 – 3 • 1 = 12 – 3 = 9;z = 2 • 1 = 2.Відповідь: (1; 9; 2).

б) x + y + z = 12; 8х + 5y + 3z = 60; z = 12 – x – y; 8х + 5y + 3(12 – x – y) = 60; 8х + 5y + 36 – 3х – 3y = 60; 5x + 2y = 60 – 36; 5x = 2у = 24; 2у = 24 – 5x|: 2; y = 12 – 2,5x; x + 12 – 2,5x + z = 12; z = 1,5x; x = 2; y = 12 – 2,5 • 2 = 12 – 5 = 7;z = 1,5 • 2 = 3. Відповідь: (2; 7; 3).

в) x + y + z = 12;

4x + 3y + 2z = 36;

z = 12 – x – y;

4x + 3y + 2(12 – x – y) = 36;

4x + 3y + 24 – 2x – 2y = 36;

2x + y = 36 – 24;

2x + y = 12;

y = 12 – 2x;

x + 12 – 2x + z = 12;

–x + z = 0;

z = x;

x = 1;

y = 12 – 2 • 1 = 12 – 2 = 10;

z = 1.

Відповідь: (1; 10; 1).