Відповіді на вправи: 514 (1-2)

1) x + y = 6;

x² + y² = 68;

2) x – y = 6;

x² + y² = 50;

y = 6 – x;

x² + (6 – x)² – 68 = 0;

x² + 36 – 12x + x² – 68 = 0;

2x² – 12x – 32 = 0|: 2;

x² – 6x – 16 = 0;

x₁ = 8, x₂ = –2.

y = x – 6;

x² + (x – 6)² – 50 = 0;

x² + x² – 12x + 36 – 50 = 0;

2x² – 12x – 14 = 0|: 2;

x² – 6x – 7 = 0;

x₁ = 7, x₂ = –1.

x = 8;

y = 6 – 8;

x = –2;

y = 6 + 2;

x = 7;

y = 7 – 6;

x = –1;

y = –1 – 6;

x = 8;

y = –2;

x = –2;

y = 8;

x = 7;

y = 1;

x = –1;

y = –7.

Відповідь: (8; –2), (–2; 8).

Відповідь: (7; 1), (–1; –7).

4) 2x + y = 5;

x² + y² = 10;

2) x + 2y = 0;

x² + y² = 54;

y = 6 – 2x;

x² + (5 – 2x)² – 10 = 0;

x² + 25 – 20x + 4x² – 10 = 0;

5x² – 20x + 15 = 0|: 5;

x² – 4x + 3 = 0;

x₁ = 1, x₂ = 3.

x = –2y;

(–2y)² + y² – 5 = 0;

4y² + y² – 5 = 0;

5y² – 5 = 0|: 5;

y² – 5 = 0;

y₁ = –1, y₂ = 1.

x = 1;

y = 5 – 2 • 1;

x = 3;

y = 5 – 2 • 3;

x = –1;

y = –2 • (–1);

y = 1;

x = –2 • 1;

x = 1;

y = 3;

x = 3;

y = –1;

x = 2;

y = –1;

x = –2;

y = 1.

Відповідь: (1; 3), (3; –2).

Відповідь: (2; –1), (–2; 1).

5) 2x = 6 + y;

2x² + y² = 66;

6) 2 + y = 3x;

3x² = 28 – y²;

y = 2x – 6;

2x² + (2x – 6)² – 66 = 0;

2x² + 4x² – 24x + 36 – 66 = 0;

6x² – 24x – 30 = 0|: 6;

x² – 4x – 5 = 0;

x₁ = 5, x₂ = –1.

y = 3x – 2;

3x² – 28 + (3x – 2)² = 0;

3x² – 28 + 9x² – 12x + 4 = 0;

12x² – 12x – 24 = 0|: 12;

x² – x – 2 = 0;

y₁ = 2, y₂ = –1.

x = 5;

y = 2 • 5 – 6;

x = –1;

y = 2 • (–1) – 6

x = 2;

y = 3 • 2 – 2;

y = –1;

x = 3 • (–1) – 2

x = 5;

y = 4;

x = –1;

y = –8.

x = 2;

y = 4;

x = –1;

y = –5.

Відповідь: (5; 4), (–1; –8).

Відповідь: (2; 4), (–1; –5).

7) 2x + y = 0;

x² + y² = 6y;

8) x – y = 7;

x² – y² = xy + 19;

y = –2x;

x² + (–2x)² – 6 • (–2x) = 0;

x² + 4x² + 12x = 0;

5x² + 12x = 0;

x(5x + 12) = 0;

x₁ = 0, x₂ = –2,4.

x = 0;

y = 0;

x = –2,4;

y = –4,8.

Відповідь: (0; 0), (–2,4; –4,8).

y = x – 7;

x² – (x – 7)² – x(x – 7) – 19 = 0;

x² – (x² – 14x + 49) – x² + 7x – 19 = 0;

–x² + 14x – 49 + 7x – 19 = 0;

–x² + 21x – 68 = 0;

D = 21² – 4 • (–1) • (–68) = 169 = 13²;

x₁ = $\frac{-21-13}{-2}$ = $\frac{-34}{-2}$ = 17;

x₂ = $\frac{-21+13}{-2}$ = $\frac{-8}{-2}$ = 4.

x = 17;

y = 17 – 7;

x = 4;

y = 4 – 7;

x = 17;

y = 19;

x = 4;

y = –3;

Відповідь: (17; 10), (4; –3).