Відповіді на вправи: 513 (1-2)

1) 2x + y = 6;

x² + xy = 8;

y = 6 – 2x;

x² = x(6 – 2x) – 8 = 0;

x² + 6x – 2x² – 8 = 0;

–x² + 6x – 8 = 0;

D = 6² – 4 • (–1) • (–8) = 4 = 2²

x₁ = $\frac{-6-2}{-2}$ = $\frac{-8}{-2}$ = 4;

x₂ = $\frac{-6+2}{-2}$ = $\frac{-4}{-2}$ = 2.

x = 4; x = 2;

y = 6 – 2 • 4; y = 6 – 2 • 2;

x = 4; x = 2;

y = –2; y = 2.

Відповідь: (4; –2), (2; 2).

2) y – 3x = 7;

x² + xy = 2;

y = 3x + 7;

x² + x(3x + 7) – 2 = 0;

x² + 3x² – 7x – 2 = 0;

4x² + 7x – 2 = 0;

D = 7² – 4 • 4 • (–2) = 81 = 9²;

x₁ = $\frac{-7-9}{8}$ = $\frac{-16}{8}$ = –2;

x₂ = $\frac{-7+9}{8}$ = $\frac{2}{8}$ = $\frac{1}{4}$.

x = –2; x = $\frac{1}{4}$;

y = 3 • (–2 + 7); y = 3 • $\frac{1}{4}$ + 7;

x = –2; x = $\frac{1}{4}$;

y = 1; y = 7$\frac{3}{4}$.

Відповідь: (–2; 1), ($\frac{1}{4}$; 7$\frac{3}{4}$).

3) 2x = y – 3;

15x2 = 2xy + 5;

y = 2x + 3;

15x² – 2x(2x + 3) – 5 = 0;

15x² – 4x² – 6x – 5 = 0;

11x² – 6x – 5 = 0;

D = (–6)² – 4 • 11 •(–5) = 256 = 16²;

x₁ = $\frac{6-16}{22}$ = $\frac{-10}{22}$ = –$\frac{5}{11}$;

x₂ = $\frac{6+16}{22}$ = $\frac{22}{22}$ = 1.

4) 4x = 2 – y;

2x² – 44 = 3xy;

y = 2 – 4x;

2x² – 44 – 3x(2 – 4x) = 0;

2x² – 44 – 6x + 12x² = 0;

14x² – 6x – 44 = 0|: 2

7x² – 3x – 22 = 0

D = (–3)² –4 • 7 • (–22) = 625 = 25²;

x₁ = $\frac{3-25}{14}$ = $\frac{-22}{14}$ = –$\frac{11}{7}$;

x₂ = $\frac{3+25}{14}$ = $\frac{28}{14}$ = 2.

x = –$\frac{5}{11}$;

y = 2•(–$\frac{5}{11}$) + 3;

x = 1;

y = 2 • 1 + 3;

x = –$\frac{11}{7}$

y = 2 – 4• (–$\frac{11}{7}$);

x = 2;

y = 2 – 4 • 2;

x = –$\frac{5}{11}$;

y = 2$\frac{1}{11}$;

x = 1;

y = 5.

x = –1$\frac{4}{7}$;

y = 8$\frac{2}{7}$;

x = 2;

y = –6.

Відповідь: (–$\frac{5}{11}$; 2$\frac{1}{11}$), (1; 5).

Відповідь: (–1$\frac{4}{7}$; 8$\frac{2}{7}$), (2; –6).