Відповіді на вправи: 13.21 (1-2)

13.21. Розв’яжіть систему рівнянь:

1) x² + 10xy + 25y² = 49, (x + 5y)² = 49, x + 5y = ±7,

x – 5y = –3; x – 5y = –3; x – 5y = –3.

a) x + 5y = –7, +

x – 5y = –3; –5 – 5y = –3; –5y = 2; y = –0,4.

2x = –10, x = –5;

б) x + 5y = 7, +

x – 5y = –3; 2 – 5y = –3; –5y = –5; y = 1.

2x = 4, x = 2;

Відповідь: (–5; –0,4), (2; 1).

2) x² + 4xy + 4y² = 4x + 2y, (x + 2y)² = 4x + 2y, 4x + 2y = 16,_

x + 2y = 4; x + 2y = 4; x + 2y = 4;

3x = 12, x = 4.

4 + 2y = 4; 2y = 0; y = 0.

Відповідь: (4; 0).

3) x² + y² = 10, x² + y² = 10, +

xy = 3; • 2 2xy = 6;

(x + y)² = 16;

x + y = 4 або x + y = –4.

a) x + y = 4, x = 4 – y, x = 4 – y, x = 4 – y,

xy = 3; y(4 – y) = 3; y² – 4y + 3 = 0; y = 3 або y = 1;

y = 3, або y = 1,

x = 1 x = 3.

б) x + y = –4, x = –y – 4, x = –y – 4, x = –y – 4,

xy = 3; xy = 3; y(–y – 4) = 3; y² – 4y + 3 = 0;

x = –y – 4, y = –3, або y = –1,

y = –3 або у = –1; х = –1 х = –3.

Відповідь: (1; 3), (3; 1), (–1; –3), (–3; –1).

4) x² + 25y² = 104, x² + 25y² = 104, +

xy = –4; • 10 10xy = –40;

(x + 5y)2 = 64;

x + 5y = 8 або x + 5y = –8.

a) x + 5y = 8, x = 8 – 5y, x = 8 – 5y,

xy = –4; y(8 – 5y) = –4; 5y² – 8y – 4 = 0.

5y² – 8y – 4 = 0; D = 64 + 80 = 144;

y₁ = $\frac{8+12}{10}$ = 2; y₂ = $\frac{8-12}{10}$ = –$\frac{4}{10}$ = –0,4.

x₁ = 8 – 5 • 2 = –2; x₂ = 8 – 5 • (–0,4) = 8 + 2 = 10.

б) x + 5y = –8, x = –5y – 8, x = –5y – 8,

xy = –4; y(–5y – 8) + 4 = 0; 5y² + 8y – 4 = 0.

5y² + 8y – 4 = 0; D = 64 + 80 = 144;

y₁ = $\frac{-8-12}{10}$ = –2; y₂ = $\frac{-8+12}{10}$ = 0,4.

x₁ = –5 • (–2) – 8 = 2; x₂ = –5 • 0,4 – 8 = –10.

Відповідь: (–2; 2), (10; –0,4); (2; –2), (–10; 0,4).