Відповіді на вправи: 24.61 (1)

24.61. Розв’яжіть систему рівнянь.

1) x – 4y = –6, x = 4y – 6,

x² + 4y² = 8; (4y – 6)² + 4y² = 8;

x = 4y – 6,

16y² – 48y + 36+ 4y² –8 = 0;

x = 4y – 6, x = 4y – 6,

20y² – 48y + 28 = 0; 5y² – 12y + 7 = 0.

5y² – 12y + 7 = 0; D = 144 – 140 = 4; y₁ = $\frac{12-2}{10}$ = 1,

y₁ = $\frac{12+2}{10}$ = 1,4; x₁ = 4 – 6 = –2, x₂ = 4 • 1,4 – 6 = –0,4.

Відповідь: (–2; 1), (–0,4; 4).

2) 3х + y = –2, y = –3x – 2,

3x² – 2xy = 28; 3x² – 2x(–3x – 2) – 28 = 0;

y = –3x – 2, y = –3x – 2,

3x² +6x² + 4x –28 = 0; 9x² + 4x – 28 = 0.

9x² + 4x – 28 = 0; D = 16 + 1008 = 1024 = 32²;

x₁ = $\frac{-4-32}{18}$ = –2; x₂ = $\frac{-4+32}{18}$ = $\frac{28}{18}$ = $\frac{14}{9}$.

y₁ = –3 • (–1) – 2 = 4; y₂ = (–3) • $\frac{14}{9}$ – 2 = –$\frac{14}{9}$ – 2 = –$\frac{20}{3}$.

Відповідь: (–2; 4), ($\frac{14}{9}$; –$\frac{20}{3}$).

3) x + 2y = 13, x = 13 – 2y, x = 13 – 2y,

xy = 15; y(13 – 2y) – 15 = 0; –2y² + 13y – 15 = 0.

2y² – 13y + 15 = 0, D = 169 – 120 = 49;

y₁ = $\frac{13-7}{4}$ = $\frac{6}{4}$ = $\frac{3}{2}$; y₂ = $\frac{13+7}{4}$ = $\frac{20}{4}$ = 5. x₁ = 13 – 2 • $\frac{3}{2}$ = 10;

x₂ = 13 – 2 • 5 = 3.

Відповідь: (10; $\frac{3}{2}$), (3; 5).

4) 3х – 2y = 18, 2у = 3х – 18, y = 1,5x – 9,

ху = –12; ху = –12; x(1,5x – 9) + 12 = 0;

y = 1,5x – 9, y = 1,5x – 9, y = 1,5x – 9,

1,5 x² – 9x + 12 = 0; x² – 6x + 8 = 0; x = 4 або x = 2;

x = 4, або x = 2,

y = –3 y = –6.

Відповідь: (4; –3), (2; –6).

5) x² – y² = 21, (x – y)(x + y) = 21, –3х – y = 21, –x + y = 7, +

x + y = –3; x + y = –3; x + y = –3; x + y = –3;

2у = 4; y = 2; x + 2 = –3; x = –5.

Відповідь: (–5; 2).

6) x² – ху + y² = 7, (x – y)² + ху = 7, ху = 6, y(y + 1) = 6,

x – y = 1; x – y = 1; x = y + 1; x = y + 1;

y² + y – 6 = 0, y = –3, або y = 2

x = y + 1; x = –2 x = 3.

Відповідь: (–2; –3), (3; 2).

7) (x – 1)(y – 1) = –2, (1 – y – 1)(y – 1) = –2, y² – y – 2 = 0,

x + y = 1; x = 1 – y; x = 1 – y;

y = 2 або y = –1, y = 2, або y = –1,

x = 1 – y; x = –1 x = 2.

Відповідь: (–1; 2), (2; –1).

8) (x – 2)(y + 1) = 1, (3 + y – 2)(y + 1) = 1, (y + 1)² = 1,

x – y = 3; x = 3 + y; x = 3 + y;

y + 1 = –1, або y + 1 = 1, y = –2 або y = 0,

x = 3 + y x = 3 + y; x = 1 x = 3.

Відповідь: (1; –2), (3; 0).

9) $\frac{1}{х}$ + $\frac{1}{у}$ = $\frac{3}{8}$, $\frac{х+у}{\mathrm{ху}}$ = $\frac{3}{8}$, $\frac{12}{\mathrm{ху}}$ = $\frac{3}{8}$, 3xy = 96, ху = 32,

x + y = 12; x + y = 12; x = 12 – y; x = 12 – y; x = 12 – y;

y(12 – y) = 32, –y² + 12y – 32 = 0, y = 8 або y = 4, y = 8, або

x = 12 – y; x = 12 – y; x = 12 – y; x = 4

y = 4,

x = 8.

Відповідь: (4; 8), (8; 4).

10) $\frac{1}{х}$ – $\frac{1}{у}$ = $\frac{4}{5}$, $\frac{у-х}{\mathrm{ху}}$ = $\frac{4}{5}$, $\frac{4}{\mathrm{ху}}$ = $\frac{4}{5}$, ху = 5, x(x + 4) = 5,

y – x = 4; y – x = 4; y = x + 4; y = x + 4; y = x + 4;

x² + 4x – 5 = 0, x = –5 або x = 1, x = –5, або x = 1,

y = x + 4; y = x + 4; y = –1 y = 5.

Відповідь: (–5; –1), (1; 5).