Відповіді на вправи: 521 (1)

521. Розв’яжіть систему рівнянь:

1) x² + y² + x + y = 18;

x² + y² + xy = 12;

(x² + 2xy + y²) – 2xy + x + y = 18;

(x² + 2xy + y²) – 2xy + xy = 12;

(x + y)² – 2xy + (x + y) = 18;

(x + y)² – xy = 12;

/x + y = a, xy = b/;

a² – 2b + a = 18;

a² – b = 12;

b = a² – 12;

a² – 2(a² – 12) + a – 18 = 0;

a² – 2a² + 24 + a – 18 = 0;

–a² + a + 6 = 0 |• (–1);

a² – a – 6 = 0;

a₁ = –2, a₂ = 3;

a = –2; a = 3;

b = (–2)² – 12; b = 3² – 12;

a = –2; a = 3;

b = –8; b = –3;

x + y = –2; x + y = 3;

xy = –8; xy = –3;

x = 2; x = –4; t² – 3t – 3 = 0;

y = –4; y = 2; D = (–3)² – 4 • 1 • (–3) = 21;

t₁ = $\frac{3-\sqrt{21}}{2}$; t₂ = $\frac{3+\sqrt{21}}{2}$;

x = $\frac{3-\sqrt{21}}{2}$; x = $\frac{3+\sqrt{21}}{2}$;

y = $\frac{3+\sqrt{21}}{2}$; y = $\frac{3-\sqrt{21}}{2}$.

Відповідь: ($\frac{3-\sqrt{21}}{2}$; $\frac{3+\sqrt{21}}{2}$), ($\frac{3+\sqrt{21}}{2}$; $\frac{3-\sqrt{21}}{2}$), (2; –4), (–4; 2).

2) x – y – xy = –1; x – y – xy = –1;

x² + y² – xy = 3; (x² – 2xy + y²) + 2xy – xy = 3;

x – y – xy = –1;

(x – y)² + xy = 3;

/x – y = a; xy = b/

a – b = –1;

a² + b = 3;

b = a + 1;

a² + a + 1 = 3;

a² + a – 2 = 0;

a₁ = –2, a₂ = 1.

a = –2; a = 1;

b = –2 + 1; b = 1 + 1;

a = –2; a = 1;

b = –1; b = 2;

x – y = –2; x – y = 1;

xy = –1; xy = 2.

y = x + 2; y = x – 1;

x(x + 2) = –1; x(x – 1) = 2;

x² + 2x + 1 = 0; x² – x – 2 = 0;

(x + 1)² = 0; x₁ = –1, x₂ = 2;

x = –1; x = –1; x = 2;

y = 1; y = –2; y = 1.

Відповідь: (–1; 1), (–1; –2), (2; 1).

3) x³ + y³ = 65; (x + y)(x² – xy + y²) = 65;

x + y = 5; x + y = 5;

5(x² – xy + y²) = 65;

x + y = 5;

x² – xy + y² = 13;

x + y = 5;

x = 5 – y;

(5 – y)² + (5 – y)y + y² – 13 = 0;

25 – 10y + y² – 5y + y² + y² – 13 = 0;

3y² – 15y + 12 = 0| : 3;

y² – 5y + 4 = 0;

y₁ = 1, y₂ = 4;

y = 1; y = 4;

x = 5 – 1; x = 5 – 4;

y = 1; y = 4;

x = 4; x = 1.

Відповідь: (4; 1), (1; 4).

4) x³ – y³ = 28; (x – y)(x² + xy + y²) = 28;

x² + y² + xy = 7; x² + y² + xy = 7;

(x – y) • 7 = 28; x – y = 4;

x² + y² + xy = 7; x² + y² + xy = 7;

x = 4 + y;

(4 + y)² + y² + (4 + y)y = 7;

16 + 8y + y² + y² + 4y + y² – 7 = 0;

3y² + 12y + 9 = 0|: 3;

y² + 4y + 3 = 0;

y₁ = –1, y₂ = –3.

y = –1; y = –3;

x = 4 – 1; x = 4 – 3;

y = –1; y = –3;

x = 3; x = 1;

Відповідь: (3; –1), (1; –3).