§8. Неравенства с двумя переменными и их системы — Дополнительные упражнения к параграфу 7 — 497 — стр. 146

$\{\begin{array}{l}(x+y)(x-y)=0\\ 2x-y=1\end{array}$

$1)\{\begin{array}{l}x+y=0\\ 2x-y=1\end{array}$

$\{\begin{array}{l}x=-y\\ -2y-y=1\end{array}$

$\{\begin{array}{l}x=-y\\ -3y=1\end{array}$

$\{\begin{array}{l}y=-\frac{1}{3}\\ x=\frac{1}{3}\end{array}$

$2)\{\begin{array}{l}x-y=0\\ 2x-y=1\end{array}$

$\{\begin{array}{l}y=x\\ 2x-x=1\end{array}$

$\{\begin{array}{l}x=1\\ y=1\end{array}$

Ответ: $(\frac{1}{3};-\frac{1}{3}),(1;1)$.

$\{\begin{array}{l}{x}^2+y^2=100\\ (x-7y)(x+7y)=0\end{array}$

$1)\{\begin{array}{l}x-7y=0\\ x^2+y^2=100\end{array}$

$\{\begin{array}{l}x=7y\\ 49y^2+y^2=100\end{array}$

$\{\begin{array}{l}{y}^2=2\\ x=7y\end{array}$

$\{\begin{array}{l}y=\sqrt{2}\\ x=7\sqrt{2}\end{array}$ или $\{\begin{array}{l}y=-\sqrt{2}\\ x=-7\sqrt{2}\end{array}$

$2)\{\begin{array}{l}x+7y=0\\ x^2+y^2=100\end{array}$

$\{\begin{array}{l}x=-7y\\ 49y^2+y^2=100\end{array}$

$\{\begin{array}{l}{y}^2=2\\ x=-7y^2\end{array}$

$\{\begin{array}{l}y=\sqrt{2}\\ x=-7\sqrt{2}\end{array}$ или $\{\begin{array}{l}y=-\sqrt{2}\\ x=7\sqrt{2}\end{array}$

Ответ: $(7\sqrt{2};\sqrt{2}),(-7\sqrt{2};-\sqrt{2}),(-7\sqrt{2};\sqrt{2}),(7\sqrt{2};-\sqrt{2})$.

$\{\begin{array}{l}{x}^2+y^2=25\\ (x-3)(y-5)=0\end{array}$

$1)\{\begin{array}{l}{x}^2+y^2=25\\ x-3=0\end{array}$

$\{\begin{array}{l}x=3\\ 9+y^2=25\end{array}$

$\{\begin{array}{l}{y}^2=16\\ x=3\end{array}$

$\{\begin{array}{l}x=3\\ y=4\end{array}$ или $\{\begin{array}{l}x=3\\ y=-4\end{array}$

$2)\{\begin{array}{l}{x}^2+y^2=25\\ y-5=0\end{array}$

$\{\begin{array}{l}y=5\\ x^2+25=25\end{array}$

$\{\begin{array}{l}y=5\\ x=0\end{array}$

Ответ: $(3;4),(3;-4),(0;5)$.

$\{\begin{array}{l}{x}^2-y^2=50\\ x(y+1)=0\end{array}$

$1)\{\begin{array}{l}x=0\\ x^2-y^2=50\end{array}$

$\{\begin{array}{l}x=0\\ -y^2=50\end{array}$ - решений нет

$2)\{\begin{array}{l}y+1=0\\ x^2-y^2=50\end{array}$

$\{\begin{array}{l}y=-1\\ x^2-1=50\end{array}$

$\{\begin{array}{l}y=-1\\ x=\sqrt{51}\end{array}$ или $\{\begin{array}{l}y=-1\\ x=-\sqrt{51}\end{array}$

Ответ: $(\sqrt{51};-1),(-\sqrt{51};-1)$.