Урок 12. Сложные системы уравнений

Домашнее задание из В.В. Ткачук “Математика – абитуриенту”

1. \(\left\{\begin{array}{l l} x+y+xy=7,\\x^2+y^2+xy=13\end{array}\right.\)
2. \(\left\{\begin{array}{l l} 2x^2+xy-y^2=20,\\x^2-4xy+7y^2=13\end{array}\right.\)
3. \(\left\{\begin{array}{l l} 3x^2+5xy-2y^2=20,\\x^2+xy+y^2=7\end{array}\right.\)
4. \(\left\{\begin{array}{l l} x^2-y^2+3y=0,\\x^2+3xy+2y^2+2x+4y=0\end{array}\right.\)
5. \(\left\{\begin{array}{l l} x^2+y^2+z=2,\\x^2+y+z^2=2,\\x+y^2+z^2=2\end{array}\right.\)
6. \(\left\{\begin{array}{l l} x^2+2yz=1,\\y^2-2zx=2,\\z^2+2xy=1\end{array}\right.\)
7. \(\left\{\begin{array}{l l} x^2+y=2,\\x+y^2=2\end{array}\right.\)
8. \(\left\{\begin{array}{l l} (x^2+1)(y^2+1)=10,\\ (x+y)(x-y)=3\end{array}\right.\)
9. \(\left\{\begin{array}{l l} xy(x+y)=20,\\\frac{1}{x}+\frac{1}{y}=\frac{5}{4}\end{array}\right.\)
10. \(\left\{\begin{array}{l l} x^4+y^4=17(x+y)^2,\\xy=2(x+y)\end{array}\right.\)
11. \(\left\{\begin{array}{l l} x^2+3=2xy,\\6x^2-11y^2=10\end{array}\right.\)
12. \(\left\{\begin{array}{l l} 2x=\frac{y}{1+y^2},\\2y=\frac{x}{1+x^2}\end{array}\right.\)
13. \(\left\{\begin{array}{l l} \frac{2}{2x-y}+\frac{3}{x-2y}=\frac{1}{2},\\\frac{2}{2x-y}-\frac{1}{x-2y}=\frac{1}{18}\end{array}\right.\)
14. \(\left\{\begin{array}{l l} x^3-\sqrt{y}=1,\\5x^6-8x^3\cdot\sqrt{y}+2y=2\end{array}\right.\)
15. \(\left\{\begin{array}{l l} 2x+y=2,\\2(y-1)=\sqrt{10x^2-xy-2y^2}\end{array}\right.\)
16. \(\left\{\begin{array}{l l} 10x^2+5y^2-2xy-38x-6y+41=0,\\3x^2-2y^2+5xy-17x-6y+20=0\end{array}\right.\)
17. \(\left\{\begin{array}{l l} y^2-4xy+4y-1=0,\\3x^2-2xy-1=0\end{array}\right.\)
18. \(\left\{\begin{array}{l l} x^2y^2-2x+y^2=0,\\2x^2-4x+3+y^3=0\end{array}\right.\)

Ответы к домашнему заданию урока 12 из В.В. Ткачук “Математика – абитуриенту”

1. (1;3), (3;1)
2. (3;2), (-3;-2), $$(-17/(2\sqrt{7}); 1/(2\sqrt{7})), (17/(2\sqrt{7}); -1/(2\sqrt{7}))$$
3. (2;1), (-2;-1), $$(-17/\sqrt{39}; 1/\sqrt{39}), (17/\sqrt{39}; -1/\sqrt{39})$$
4. (0;0), (2;-1), (-10/7; -4/7)
5. $$((-1+\sqrt{17})/4; (-1+\sqrt{17})/4; (-1+\sqrt{17})/4)$$, $$((-1-\sqrt{17})/4; (-1-\sqrt{17})/4; (-1-\sqrt{17})/4)$$, (0;1;1), (1;0;1), (1;1;0), (3/2; -1/2; -1/2), (-1/2; 3/2; -1/2), (-1/2;-1/2; 3/2)
6. $$((\sqrt{2}-1)/\sqrt{2\sqrt{2}-1}; 2/\sqrt{2\sqrt{2}-1}; (\sqrt{2}-1)/\sqrt{2\sqrt{2}-1})$$, $$(-(\sqrt{2}-1)/\sqrt{2\sqrt{2}-1};-2/\sqrt{2\sqrt{2}-1};-(\sqrt{2}-1)/\sqrt{2\sqrt{2}-1})$$, (1;0;1), (-1;0;-1)
7. (1;1), (-2;-2), $$((1+\sqrt{5})/2; (1-\sqrt{5})/2)$$, $$((1-\sqrt{5})/2;(1+\sqrt{5})/2)$$
8. (2;1), (-2;1), (2;-1), (-2;-1)
9. (1;4), (4;1), $$((-5-\sqrt{41})/2;(-5+\sqrt{41})/2)$$, $$((-5+\sqrt{41})/2;(-5-\sqrt{41})/2)$$
10. (0;0), (-2;1), (1;-2), (6;3), (3;6)
11. (3;2), (-3;-2)
12. (0;0)
13. (5;-2)
14. $$(\sqrt[3]{4}; 9)$$
15. (1/2; 1)
16. (2;1)
17. (1;1), $$((2\sqrt{6}-3)/15; -(4\sqrt{6}+9)/5)$$, $$(-(2\sqrt{6}+3)/15; (4\sqrt{6}-9)/5)$$
18. (1;-1)