Відповіді на вправи: 512 (1-2)

1) x – y = 2;

2xy – y² = –3;

x = 2 + y;

2(2 + y)y – y² + 3 = 0;

4y + 2y² – y² + 3 = 0;

y² + 4y + 3 = 0;

y₁ = –1, y₂ = –3.

y = –1; y = –3;

x – (–1) = 2; x – (–3) = 2;

y = –1; y = –3.

x = 1; x = –1.

Відповідь: (1; –1), (–1; –3).

3) x + 2y = 4;

y² – 4xy = –7;

x = 4 – 2y;

y² – 4(4 – 2y)y = –7;

y² – 16y + 8y² + 7 = 0;

9y² – 16y + 7 = 0;

D = (–16)² – 4 • 9 • 7 = 4 = 2²;

y₁ = $\frac{16-2}{18}$ = $\frac{14}{18}$ = $\frac{7}{9}$;

y₂ = $\frac{16+2}{18}$ = $\frac{18}{18}$ = 1.

y = $\frac{7}{9}$; y = 1;

x + 2 • $\frac{7}{9}$ = 4; x + 2 • 1 = 4;

y = $\frac{7}{9}$; y = 1.

x = 2$\frac{4}{9}$; x = 2.

Відповідь: (2$\frac{4}{9}$; $\frac{7}{9}$), (2; 1).

2) x – 3y = –10;

xy – y² = –12;

x = 3y – 10;

(3y – 10)y – y² = –12;

3y² – 10y – y² + 12 = 0;

2y² – 10y + 12 = 0| : 2;

y² – 5y + 6 = 0;

y₁ = 2, y₂ = 3;

y = 2 ; y = 3

x – 3 • 2 = –10; x – 3 • 3 = –10;

y = 2; y = 3.

x = –4; x = –1.

Відповідь: (–4; 2), (–1; 3).

4) x + y = 5;

2xy = y² – 13;

x = 5 – y;

2(5 – y)y – y² + 13 = 0;

10y – 2y² – y² + 13 = – 3y² + 10y + 13 = 0;

D = 10² – 4 • (–3) •13 = 256 = 16²;

y₁ = $\frac{-10-16}{-6}$ = $\frac{-26}{-6}$ = $\frac{13}{3}$ = 4$\frac{1}{3}$;

y₂ = $\frac{-10+16}{-6}$ = $\frac{6}{-6}$ = –1.

y = 4$\frac{1}{3}$; y = –1;

x = 5 – 4$\frac{1}{3}$; x = 5 + 1;

y = 4$\frac{1}{3}$; y = –1;

x = $\frac{2}{3}$; x = 6.

Відповідь: ($\frac{2}{3}$; 4$\frac{1}{3}$), (6; –1).

3) x + 2y = 4;

y² – 4xy = –7;

x = 4 – 2y;

y² – 4(4 – 2y)y = –7;

y² – 16y + 8y² + 7 = 0;

9y² – 16y + 7 = 0;

D = (–16)² – 4 • 9 • 7 = 4 = 2²;

y₁ = $\frac{16-2}{18}$ = $\frac{14}{18}$ = $\frac{7}{9}$;

y₂ = $\frac{16+2}{18}$ = $\frac{18}{18}$ = 1.

y = $\frac{7}{9}$; y = 1;

x + 2 • $\frac{7}{9}$ = 4; x + 2 • 1 = 4;

y = $\frac{7}{9}$; y = 1.

x = 2$\frac{4}{9}$; x = 2.

Відповідь: (2$\frac{4}{9}$; $\frac{7}{9}$), (2; 1).

4) x + y = 5;

2xy = y² – 13;

x = 5 – y;

2(5 – y)y – y² + 13 = 0;

10y – 2y² – y² + 13 = – 3y² + 10y + 13 = 0;

D = 10² – 4 • (–3) •13 = 256 = 16²;

y₁ = $\frac{-10-16}{-6}$ = $\frac{-26}{-6}$ = $\frac{13}{3}$ = 4$\frac{1}{3}$;

y₂ = $\frac{-10+16}{-6}$ = $\frac{6}{-6}$ = –1.

y = 4$\frac{1}{3}$; y = –1;

x = 5 – 4$\frac{1}{3}$; x = 5 + 1;

y = 4$\frac{1}{3}$; y = –1;

x = $\frac{2}{3}$; x = 6.

Відповідь: ($\frac{2}{3}$; 4$\frac{1}{3}$), (6; –1).

5) x + y = 4;

x² + 16 = 5xy;

x = 4 – y;

(4 – y)² + 16 – 5(4 – y)y = 0;

16 – 8y + y² + 16 – 20y + 5y² = 0;

6y² – 28y + 32 = 0|: 2;

3y² – 14y + 16 = 0;

D = (–14)² – 4 • 3 • 16 = 4 = 2²;

y₁ = $\frac{14-2}{6}$ = $\frac{12}{6}$ = 2;

y₂ = $\frac{14+2}{6}$ = $\frac{16}{6}$ = $\frac{8}{3}$ = 2$\frac{2}{3}$;

y = 2$\frac{2}{3}$ y = 2;

x = 4 – 2$\frac{2}{3}$ x = 4 – 2;

y = 2$\frac{2}{3}$; y = 2.

x = 1$\frac{1}{3}$; x = 2.

Відповідь: (1$\frac{1}{3}$; 2$\frac{2}{3}$), (2; 2).

6) 3x = 5 – y;

3xy = 2x² + 4;

y = 5 – 3x;

3x(5 – 3x) – 2x² – 4 = 0;

15x – 9x² – 2x² – 4 = 0;

–11x² + 15x – 4 = 0;

D = 15² – 4 • (–11) • (–4) = 49 = 7²;

x₁ = $\frac{-15-7}{-22}$ = $\frac{-22}{-22}$ = 1;

x₂ = $\frac{-15+7}{-22}$ = $\frac{-8}{-22}$ = $\frac{4}{11}$;

x = 1; x = $\frac{4}{11}$;

y = 5 – 3 • 1; y = 5 – 3 • $\frac{4}{11}$;

x = 1; x = $\frac{4}{11}$;

y = 2; y = 3$\frac{10}{11}$.

Відповідь: (1; 2), ($\frac{4}{11}$; 3$\frac{10}{11}$).

7) 2у = 7 – x;

2y² = 14 – xy;

x = 7 – 2y;

2y² – 14 + (7 – 2y)y = 0;

2y² – 14 + 7y – 2y² = 0;

7y = 14;

y = 2;

y = 2;

x = 7 – 2 • 2;

y = 2;

x = 3.

Відповідь: (3; 2).

8) 3y = –x – 2;

2xy = 3y2 – 5;

x + 2 = 3y;

2xy = 3y2 – 5;

x = –3y – 2;

2(–3y – 2)y – 3y² + 5 = 0;

–6y² – 4y – 3y² + 5 = 0;

–9y² – 4y + 5 = 0;

D = (–4)² – 4 • (–9) • 5 = 196 = 14²;

y₁ = $\frac{4-14}{-18}$ = $\frac{-10}{-18}$ = $\frac{5}{9}$;

y₂ = $\frac{4+14}{-18}$ = $\frac{18}{-18}$ = –1.

y = $\frac{5}{9}$; y = –1;

x = –3 • $\frac{5}{9}$ – 2 x = –3 • (–1) – 2

y = $\frac{5}{9}$; y = –1.

x = –3$\frac{2}{3}$; x = 1.

Відповідь: (–3$\frac{2}{3}$; $\frac{5}{9}$), (1; –1).