Příklad 1

V množině $\mathbb R^2$ řešte soustavy

(a) $\begin{cases}x+y=2\\ x+y^2=4\end{cases}$	(b) $\begin{cases}4y-3=6\\ 2y^2-3x=0\end{cases}$
(c) $\begin{cases}y=2x+3\\ y=x^2-x-1\end{cases}$	(d) $\begin{cases}3y-4x=7\\ y^2-4y-3x+1\end{cases}$
(e) $\begin{cases}x=2y+1\\ y=x^2-1\end{cases}$	(f) $\begin{cases}x-3y=10\\ x^2-24y=100\end{cases}$

Řešení	Ukázat
(a) $[x;y]\in\{[0;2];[3;-1]\}$ (b) $[x;y]\in\{[6;3]\}$ (c) $[x;y]\in\{[-1;1];[4;11]\}$ (d) $[x;y]\in\{[-\frac{13}{16};\frac54];[2;5]\}$ (e) $[x;y]\in\{[-\frac12;-\frac34];[1;0]\}$ (f) $[x;y]\in\{[-2;-4];[10;0]\}$

Příklad 2

V množině $\mathbb R^2$ řešte soustavy

(a $\begin{cases}x+y=1\\ x^2+y^2=5\end{cases}$	(b) $\begin{cases}x+y=5\\ x^2+y^2=17\end{cases}$
(c) $\begin{cases}x-y=0\\ x^2+y^2=10\end{cases}$	(d) $\begin{cases}5x+y=7\\ 4x^2+3y^2=16\end{cases}$
(e) $\begin{cases}x-3y=-2\\ 3x^2-7y^2=12\end{cases}$	(f) $\begin{cases}x^2+y^2+6x+2y=0\\x+y=-8\end{cases}$
(g) $\begin{cases}\|x\|+\|y\|=2\\ x^2-y=2\end{cases}$	(h) $\begin{cases}\|x\|+\|y\|=3\\ x^2+y^2=5\end{cases}$
(i) $\begin{cases}y=12x^2-5\|x\|-36\\y=6x^2-5x-12\end{cases}$	(j) $\begin{cases}x-\dfrac{x-y}2=4\\ y-\dfrac{x+3y}{x+2}=1\end{cases}$

Řešení	Ukázat
(a) $[x;y]\in\{[-1;2];[2;-1]\}$ (b) $[x;y]\in\{[1;4];[4;1]\}$ (c) $[x;y]=\{[-\sqrt5;-\sqrt5];[\sqrt5;\sqrt5]\}$ (d) $[x;y]\in\{[\frac{131}{79};-\frac{102}{79}];[1;2]\}$ (e) $[x;y]\in\{[-2;0];[\frac{17}5;\frac95]\}$ (f) $[x;y]\in\{[-6;-2];[-4;-4]\}$ (g) $[x;y]\in\{[0;-2];[\pm1;-1]\}$ (h) $[x;y]\in\{[-2;\pm1];[-1;\pm2];[1;\pm2];[2;\pm1]\}$ (i) $[x;y]\in\{[2;2];[-3;57]\}$ (j) $[x;y]=\{[2;6];[5;3]\}$

Příklad 3

Rešte pro $m,n\in\mathbb{Z}^2$ soustavu

$\begin{cases}5n-3m=15\\n^2+m^2\le16\end{cases}$

Řešení	Ukázat
$[m;n]=[0;3]$

Příklad 4

V množině $\mathbb R^2$ řešte soustavy

(a) $\begin{cases}x+y=4\\ xy=-12\end{cases}$	(b) $\begin{cases}x+y=10\\ xy=29\end{cases}$
(c) $\begin{cases}x+y=15\\ xy-34\end{cases}$	(d) $\begin{cases}x-y=2\\ xy=2\end{cases}$
(e) $\begin{cases}x-y=5\\ xy=14\end{cases}$	(f) $\begin{cases}3x-4y=9\\ xy=21\end{cases}$
(g) $\begin{cases}2x-3y=-1\\ xy=35\end{cases}$	(h) $\begin{cases}3x+5y=30\\ xy=15\end{cases}$

Řešení	Ukázat
(a) $[x;y]\in\{[-2;6];[6;-2]\}$ (b) $[x;y]\in\emptyset$ (c) $[x;y]\in\{[-2;17];[17;-2]\}$ (d) $[x;y]\in\{[1-\sqrt3;-1-\sqrt];[1+\sqrt3;\sqrt3-1]\}$ (e) $[x;y]\in\{[-2;-7];[7;2]\}$ (f) $[x;y]\in\{[7;3];[-4;-\frac{21}4]\}$ (g) $[x;y]\in\{[7;5];[-\frac{15}2;-\frac{14}3]\}$ (h) $[x;y]\in\{[5;3]\}$

Příklad 5

V množině $\mathbb R^2$ řešte soustavy

(a) $\begin{cases}\dfrac xy=\dfrac13\\ y^2-x^2=1800\end{cases}$	(b) $\begin{cases}x-3y=-2\\ x^2-2xy=7\end{cases}$
(c) $\begin{cases}x+xy=1\\ x=3y+1\end{cases}$	(d) $\begin{cases}x^2+xy+y^2=7\\ 3x+y=3\end{cases}$
(e) $\begin{cases}x^2-xy-y^2=7\\ x+y=5\end{cases}$	(f) $\begin{cases}3x^2-xy+y^2+x-y-6=0\\ 4x-y+6=0\end{cases}$
(g) $\begin{cases}x+y=5\\ x^2+3xy+2y^2=40\end{cases}$	(h) $\begin{cases}x-y=1\\ 2x^2-xy+3y^2-7x=12y-1\end{cases}$
(i) $\begin{cases}2x^2+15xy+4y^2+43x+24y+7=0\\ x-2y+5=0\end{cases}$	(j) $\begin{cases}x^2-3xy+4y^2-6x+2y=0\\ x-2y=3\end{cases}$
(k) $\begin{cases}2x+3y=7\\ xy+y^2=5\end{cases}$	(l) $\begin{cases}7x-2y=12\\ x^2-y^2=12(x-y)\end{cases}$
(m) $\begin{cases}\dfrac1{y-1}-\dfrac1{y+1}=\dfrac1x\\ y^2-x-5=0\end{cases}$	(n) $\begin{cases}\dfrac1{y-1}-\dfrac1{y+1}=\dfrac1{x^2}\\ x^2-y-5=0\end{cases}$

Řešení	Ukázat
(a) $[x;y]\in\{[-15;-45];[15;45]\}$ (b) $[x;y]\in\{[-3;-\frac13];[7;3]\}$ (c) $[x;y]\in\{[-3;-\frac43];[1;0]\}$ (d) $[x;y]\in\{[\frac17;\frac{18}7];[2;-3]\}$ (e) $[x;y]\in\{[-\frac{5+3\sqrt{17}}2;\frac{15+3\sqrt{17}}2];[\frac{-5+3\sqrt{17}}2;\frac{15-3\sqrt{17}}2]\}$ (f) $[x;y]\in\{[-\frac85;-\frac25];[-1;2]\}$ (g) $[x;y]\in\{[2;3]\}$ (i) $[x;y]\in\{[-1;2];[-\frac{184}{21};-\frac{79}{42}]\}$ (j) $[x;y]\in\{[1;-1];[12;\frac92]\}$ (h) $[x;y]\in\{[3-\sqrt5;2-\sqrt5];[3+\sqrt5;2+\sqrt5]\}$ (k) $[x;y]\in\{[-4;5];[\frac12;2]\}$ (l) $[x;y]\in\{[4;8];[\frac{12}5;\frac{12}5]\}$ (m) $[x;y]\in\{[4;-3];[4;3]\}$ (n) $[x;y]\in\{[\pm\sqrt{6-2\sqrt3};1-2\sqrt3];[\pm\sqrt{6+2\sqrt3};1+2\sqrt3]\}$

Řešení

Ukázat

(a)

$[x;y]\in\{[-15;-45];[15;45]\}$ (b)

$[x;y]\in\{[-3;-\frac13];[7;3]\}$ (c)

$[x;y]\in\{[-3;-\frac43];[1;0]\}$ (d)

$[x;y]\in\{[\frac17;\frac{18}7];[2;-3]\}$ (e)

$[x;y]\in\{[-\frac{5+3\sqrt{17}}2;\frac{15+3\sqrt{17}}2];[\frac{-5+3\sqrt{17}}2;\frac{15-3\sqrt{17}}2]\}$ (f)

$[x;y]\in\{[-\frac85;-\frac25];[-1;2]\}$ (g)

$[x;y]\in\{[2;3]\}$
(i)

$[x;y]\in\{[-1;2];[-\frac{184}{21};-\frac{79}{42}]\}$ (j)

$[x;y]\in\{[1;-1];[12;\frac92]\}$
(h)

$[x;y]\in\{[3-\sqrt5;2-\sqrt5];[3+\sqrt5;2+\sqrt5]\}$ (k)

$[x;y]\in\{[-4;5];[\frac12;2]\}$ (l)

$[x;y]\in\{[4;8];[\frac{12}5;\frac{12}5]\}$
(m)

$[x;y]\in\{[4;-3];[4;3]\}$ (n)

$[x;y]\in\{[\pm\sqrt{6-2\sqrt3};1-2\sqrt3];[\pm\sqrt{6+2\sqrt3};1+2\sqrt3]\}$

Příklad 6

V množině $\mathbb R^2$ řešte soustavy

(a) $\begin{cases} 4x^2+9y^2=72\\ 3x^2-2y^2=19\end{cases}$	(b) $\begin{cases} 3x^2-y^2=27\\x^2-y^2=-45\end{cases}$
(c) $\begin{cases} 5x^2+3y^2=92\\2x^2+5y^2=52\end{cases}$	(d) $\begin{cases} x^2+3x^2=43\\3x^2+y^2=57\end{cases}$
(e) $\begin{cases} 9x^2+y^2=90\\ x^2+6y^2=90\end{cases}$	(f) $\begin{cases} \dfrac2{x^2}-\dfrac{3}{y^2}=5\\ \dfrac{1}{x^2}+\dfrac{2}{y^2}=6\end{cases}$
(g) $\begin{cases}x^2+2y^2=208\\ 3x^2-y^2=1\end{cases}$	(h) $\begin{cases}2x^2+y^2=6\\ x^2+y^2+2x=3 \end{cases}$
(i) $\begin{cases}4x^2-9y^2=36\\ x^2-2x+y^2=15 \end{cases}$	(j) $\begin{cases}x^2+y^2+3x+y=40\\ x^2+y^2=25 \end{cases}$
(k) $\begin{cases}x^2+2y^2=9\\ (x+1)^2+2(y+1)^2=22 \end{cases}$	(l) $\begin{cases}(x-2)^2+(y-2)^2=1\\ x^2=4-y^2 \end{cases}$
(m) $\begin{cases}x^2+y^2-2x+3y=9\\ 2x^2+2y^2+x-4y=3\end{cases}$	(n) $\begin{cases}x^2+y^2-2x+3y=9\\ 2x^2+2y^2-3x+5y=17\end{cases}$

Řešení	Ukázat
(a) $[x;y]\in\{[-3;\pm2];[3;\pm2]\}$ (b) $[x;y]\in\{[-6;\pm9];[6;\pm9]\}$ (c) $[x;y]\in\{[-4;\pm2];[2;\pm2]\}$ (d) $[x;y]\in\{[-\frac{\sqrt{43}}2;\pm\frac{3\sqrt{11}}2];[\frac{\sqrt{43}}2;\pm\frac{3\sqrt{11}}2]\}$ (e) $[x;y]\in\{[-15\sqrt{\frac2{53}};\pm12\sqrt{\frac5{53}}];[15\sqrt{\frac2{53}};\pm12\sqrt{\frac5{53}}]\}$ (f) $[x;y]\in\{[-\frac12;\pm1];[\frac12;\pm1]\}$ (g) $[x;y]\in\{[-\sqrt{30};\pm\sqrt{89}];[\sqrt{30};\pm\sqrt{89}]\}$ (h) $[x;y]\in\{[-1;\pm2]\}$ (i) $[x;y]\in\{[-3;0];[\frac{57}{13};\pm\frac{16\sqrt3}{13}\}$ (j) $[x;y]\in\{[4;3];[5;0]\}$ (k) $[x;y]\in\{[1;2];[\frac73;\frac43]\}$ (l) $[x;y]\in\{[\frac{11-\sqrt7}8;\frac{11+\sqrt7}8];[\frac{11+\sqrt7}8;\frac{11-\sqrt7}8]\}$ (m) $[x;y]\in\{[-\frac95;\frac35];[1;2]\}$ (n) $[x;y]\in\{[-\frac52;-\frac32];[1;2]\}$

Řešení

Ukázat

(a)

$[x;y]\in\{[-3;\pm2];[3;\pm2]\}$ (b)

$[x;y]\in\{[-6;\pm9];[6;\pm9]\}$ (c)

$[x;y]\in\{[-4;\pm2];[2;\pm2]\}$
(d)

$[x;y]\in\{[-\frac{\sqrt{43}}2;\pm\frac{3\sqrt{11}}2];[\frac{\sqrt{43}}2;\pm\frac{3\sqrt{11}}2]\}$ (e)

$[x;y]\in\{[-15\sqrt{\frac2{53}};\pm12\sqrt{\frac5{53}}];[15\sqrt{\frac2{53}};\pm12\sqrt{\frac5{53}}]\}$
(f)

$[x;y]\in\{[-\frac12;\pm1];[\frac12;\pm1]\}$ (g)

$[x;y]\in\{[-\sqrt{30};\pm\sqrt{89}];[\sqrt{30};\pm\sqrt{89}]\}$ (h)

$[x;y]\in\{[-1;\pm2]\}$
(i)

$[x;y]\in\{[-3;0];[\frac{57}{13};\pm\frac{16\sqrt3}{13}\}$ (j)

$[x;y]\in\{[4;3];[5;0]\}$ (k)

$[x;y]\in\{[1;2];[\frac73;\frac43]\}$
(l)

$[x;y]\in\{[\frac{11-\sqrt7}8;\frac{11+\sqrt7}8];[\frac{11+\sqrt7}8;\frac{11-\sqrt7}8]\}$ (m)

$[x;y]\in\{[-\frac95;\frac35];[1;2]\}$ (n)

$[x;y]\in\{[-\frac52;-\frac32];[1;2]\}$

Příklad 7

V množině $\mathbb R^2$ řešte soustavy

(a) $\begin{cases}x^2+y=1\\ y^2+x=1 \end{cases}$	(b) $\begin{cases}x^2+y=7\\ x+y^2=7 \end{cases}$
(c) $\begin{cases}x^2+y=20\\x+y^2=20 \end{cases}$	(d) $\begin{cases}2y=4-x^2\\ 2x=4-y^2 \end{cases}$
(e) $\begin{cases}x^2+y^2=5\\ xy=2 \end{cases}$	(f) $\begin{cases}x^2+y^2=25\\ xy=12 \end{cases}$
(g) $\begin{cases}x^2+y^2=34\\ xy=-15 \end{cases}$	(h) $\begin{cases}x^2-y^2=24\\ xy=35 \end{cases}$

Řešení	Ukázat
(a) $[x;y]\in\{[\frac{-1-\sqrt5}2;\frac{-1-\sqrt5}2];[\frac{-1+\sqrt5}2;\frac{-1+\sqrt5}2]\}$ (b) $[x;y]\in\{[\frac{-1-\sqrt{29}}2;\frac{-1-\sqrt{29}}2];[\frac{-1+\sqrt{29}}2;\frac{-1+\sqrt{29}}2]\}$ (c) $[x;y]\in\{[\frac{1-\sqrt{77}}2;\frac{1+\sqrt{77}}2];[\frac{1+\sqrt{77}}2;\frac{1-\sqrt{77}}2]\}$ (d) $[x;y]\in\{[0;2];[2;0];[-1-\sqrt5;-1-\sqrt5];[-1+\sqrt5;-1+\sqrt5]\}$ (e) Tato rovnice se dá řešit přímo, $\begin{cases}x^2+y^2=5\\ xy=2\end{cases} \Rightarrow\begin{cases}x^2+y^2=5\\ y=\dfrac2x\end{cases}\Rightarrow\begin{cases}x^2+\dfrac4{x^2}=5\\ y=\dfrac2x\end{cases}\Rightarrow\begin{cases}x^4-5x^2+4=0\\ y=\dfrac2x\end{cases}\Rightarrow\begin{cases}(x^2-4)(x^2-1)=0\\ y=\dfrac2x\end{cases}\\ \Rightarrow \left[\begin{array}{l} [x;y]=[2;1]\\ [x;y]=[-2;-1]\\ [x;y]=[1;2]\\ [x;y]=[-1;-2]\end{array}\right.$ protože v průběhu řešení vznikla jednoduchá bikvadratická rovnice. Ve složitějších případech tomu tak nebývá, a proto bývá efektivnější použít druhý postup. Zavedeme substituci $x+y=a$ , $xy=b$ $\begin{cases}x^2+y^2=5\\ xy=2 \end{cases}\Rightarrow\begin{cases}a^2-2b=5\\ b=2\end{cases}\Rightarrow\begin{cases}a^2=9\\b=2 \end{cases}\Rightarrow \left[\begin{array}{l} \begin{cases}x+y=3\\xy=2 \end{cases}\\ \begin{cases}x+y=-3\\ xy=2 \end{cases}\end{array}\right.\Rightarrow \left[\begin{array}{l} [x;y]=[2;1]\\ [x;y]=[1;2]\\ [x;y]=[-2;-1]\\ [x;y]=[-1;-2]\end{array}\right.$ (f) $[x;y]\in\{[-4;-3];[-3;-4];[3;4];[4;3]\}$ (g) $[x;y]\in\{[-5;3];[-3;5];[3;-5];[5;-3]\}$ (h) U tohoto typu soustavy pomáhá jiný postup (kromě přímého dosazení). První rovnici vynásobíme 35, druhou 24 a rovnice odečteme. Soustava tak přejde na tvar $\begin{cases}x^2-y^2=24&\|\cdot35\\-\\xy=35&\|\cdot24\end{cases}\Rightarrow\begin{cases}35x^2-24xy-35y^2=0\\xy=35\end{cases}\Rightarrow\begin{cases}(7x+5y)(5x-7y)=0\\xy=35\end{cases}\Rightarrow\\ \Rightarrow\left[\begin{array}{l}\begin{cases}y=-\frac75x\\ xy=35\end{cases}\\ \begin{cases}y=\frac57x\\ xy=35\end{cases}\end{array}\right.\Rightarrow\left[\begin{array}{l}[x;y]=[7;5]\\ [x;y]=[-7;-5]\end{array}\right.$ Další šikovný postup je substituce $y=kx$ . To lze v případě, kdy $[0;0]$ není řešením soustavy. $\begin{cases}x^2-kx^2=24\\ kx^2=35\end{cases}\Rightarrow\begin{cases}x^2=\dfrac{24}{1-k^2}&(1)\\ x^2=\dfrac{35}k&(2)\end{cases}$ Porovnáním obou rovnic získáme kvadratickou rovnici $\dfrac{24}{1-k^2}=\dfrac{35}k\\ 35k^2+24k-35=0$ jejímž řešením dostaneme $k$ , z $(2)$ pak $x$ a z úvodní substituce $y$ .

Řešení

Ukázat

(a)

$[x;y]\in\{[\frac{-1-\sqrt5}2;\frac{-1-\sqrt5}2];[\frac{-1+\sqrt5}2;\frac{-1+\sqrt5}2]\}$ (b)

$[x;y]\in\{[\frac{-1-\sqrt{29}}2;\frac{-1-\sqrt{29}}2];[\frac{-1+\sqrt{29}}2;\frac{-1+\sqrt{29}}2]\}$
(c)

$[x;y]\in\{[\frac{1-\sqrt{77}}2;\frac{1+\sqrt{77}}2];[\frac{1+\sqrt{77}}2;\frac{1-\sqrt{77}}2]\}$ (d)

$[x;y]\in\{[0;2];[2;0];[-1-\sqrt5;-1-\sqrt5];[-1+\sqrt5;-1+\sqrt5]\}$

(e) Tato rovnice se dá řešit přímo,

$\begin{cases}x^2+y^2=5\\ xy=2\end{cases} \Rightarrow\begin{cases}x^2+y^2=5\\ y=\dfrac2x\end{cases}\Rightarrow\begin{cases}x^2+\dfrac4{x^2}=5\\ y=\dfrac2x\end{cases}\Rightarrow\begin{cases}x^4-5x^2+4=0\\ y=\dfrac2x\end{cases}\Rightarrow\begin{cases}(x^2-4)(x^2-1)=0\\ y=\dfrac2x\end{cases}\\ \Rightarrow \left[\begin{array}{l} [x;y]=[2;1]\\ [x;y]=[-2;-1]\\ [x;y]=[1;2]\\ [x;y]=[-1;-2]\end{array}\right.$

protože v průběhu řešení vznikla jednoduchá bikvadratická rovnice. Ve složitějších případech tomu tak nebývá, a proto bývá efektivnější použít druhý postup.

Zavedeme substituci $x+y=a$ , $xy=b$

$\begin{cases}x^2+y^2=5\\ xy=2 \end{cases}\Rightarrow\begin{cases}a^2-2b=5\\ b=2\end{cases}\Rightarrow\begin{cases}a^2=9\\b=2 \end{cases}\Rightarrow \left[\begin{array}{l} \begin{cases}x+y=3\\xy=2 \end{cases}\\ \begin{cases}x+y=-3\\ xy=2 \end{cases}\end{array}\right.\Rightarrow \left[\begin{array}{l} [x;y]=[2;1]\\ [x;y]=[1;2]\\ [x;y]=[-2;-1]\\ [x;y]=[-1;-2]\end{array}\right.$

(f) $[x;y]\in\{[-4;-3];[-3;-4];[3;4];[4;3]\}$ (g) $[x;y]\in\{[-5;3];[-3;5];[3;-5];[5;-3]\}$

(h) U tohoto typu soustavy pomáhá jiný postup (kromě přímého dosazení). První rovnici vynásobíme 35, druhou 24 a rovnice odečteme. Soustava tak přejde na tvar

$\begin{cases}x^2-y^2=24&|\cdot35\\-\\xy=35&|\cdot24\end{cases}\Rightarrow\begin{cases}35x^2-24xy-35y^2=0\\xy=35\end{cases}\Rightarrow\begin{cases}(7x+5y)(5x-7y)=0\\xy=35\end{cases}\Rightarrow\\ \Rightarrow\left[\begin{array}{l}\begin{cases}y=-\frac75x\\ xy=35\end{cases}\\ \begin{cases}y=\frac57x\\ xy=35\end{cases}\end{array}\right.\Rightarrow\left[\begin{array}{l}[x;y]=[7;5]\\ [x;y]=[-7;-5]\end{array}\right.$

Další šikovný postup je substituce

$y=kx$ . To lze v případě, kdy

$[0;0]$ není řešením soustavy.

$\begin{cases}x^2-kx^2=24\\ kx^2=35\end{cases}\Rightarrow\begin{cases}x^2=\dfrac{24}{1-k^2}&(1)\\ x^2=\dfrac{35}k&(2)\end{cases}$

Porovnáním obou rovnic získáme kvadratickou rovnici

$\dfrac{24}{1-k^2}=\dfrac{35}k\\ 35k^2+24k-35=0$

jejímž řešením dostaneme

$k$ , z

$(2)$ pak

$x$ a z úvodní substituce

$y$ .

Příklad 8

V množině $\mathbb R^2$ řešte soustavy

(a) $\begin{cases}x+\dfrac1y=\dfrac32\\ \dfrac1x+y=3\end{cases}$	(b) $\begin{cases}x+\dfrac{1}{y}=5\\x^{2}+\dfrac{1}{y^{2}}=13\end{cases}$
(c) $\begin{cases}x+y-xy=1\\ x^2+y^2-xy=3\end{cases}$	(d) $\begin{cases}x+y+xy=7\\x^2+y^2+xy=13\end{cases}$
(e) $\begin{cases}x^2+y^2+3x+3y=8\\xy+4x+4y=2 \end{cases}$	(f) $\displaystyle \left\{\begin{array}{l} x+xy+y=-9\\x^2+y^2=17 \end{array}\right.$
(g) $\begin{cases}x-y=xy\\ 1-x-y=xy\end{cases}$	(h) $\begin{cases}x^2+y^2-2x-2y=12\\ xy=6\end{cases}$
(i) $\begin{cases}x^2+xy+y^2=4\\ x+xy+y=2\end{cases}$	(j) $\begin{cases}(x+y)xy=120\\ (x-y)xy=30\end{cases}$
(k) $\begin{cases}x(x+y)=9\\ y(x+y)=16\end{cases}$ (l)	(l) $\begin{cases}x^2+y^2=34\\ x+y+xy=23\end{cases}$
(m) $\begin{cases} x^2-xy+8=0\\x^2-8x+y=0\end{cases}$	(n) $\begin{cases}x-xy+y=1\\ x^2+y^2+2x+2y=11\end{cases}$
(o) $\begin{cases} (x+y)^2+2x=35-2y\\ (x-y)^2-2y=3-2x\end{cases}$	(p) $\begin{cases}x^2+y=y^2+x\\ y^2+x=6\end{cases}$

Řešení	Ukázat
(a) $[x;y]\in\{[\frac12;1];[1;2]\}$ (b) $[x;y]\in\{[2;\frac13];[3;\frac12\}$ (c) $[x;y]\in\{[-1;1],[1;-1];[1;2];[2;1]\}$ (d) $[x;y]\in\{[1;3];[3;1]\}$ (e) $[x;y]\in\{[-1;2];[2;-1]\}$ (f) $[x;y]\in\{[-\frac{1+\sqrt{33}}2;\frac{\sqrt{33}-1}2];[\frac{\sqrt{33}-1}2;-\frac{1+\sqrt{33}}2]\}$ (g) $[x;y]\in\{[\frac12;\frac13]\}$ (h) $[x;y]\in\{[3-\sqrt3;3+\sqrt3];[3+\sqrt3;3-\sqrt3]\}$ (i) $[x;y]\in\{[0;2];[2;0]\}$ (j) $[x;y]\in\{[5;3]\}$ (k) $[x;y]\in\{[-\frac95;-\frac{16}5];[\frac95;\frac{16}5]\}$ (l) $[x;y]\in\{[3;5];[5;3]\}$ (m) $[x;y]\in\{[-1;-9];[4\pm2\sqrt2;8]\}$ (n) $[x;y]\in\{[-4;1];[1;-4];[1;2];[2;1]\}$ (o) $[x;y]\in\{[-5;-2];[-3;-4];[1;4];[3;2]\}$ (p) $[x;y]\in\{[-3;-3];[2;2];[\frac{1-\sqrt{21}}2;\frac{1+\sqrt{21}}2];[\frac{1+\sqrt{21}}2;\frac{1-\sqrt{21}}2]\}$

Řešení

Ukázat

(a)

$[x;y]\in\{[\frac12;1];[1;2]\}$ (b)

$[x;y]\in\{[2;\frac13];[3;\frac12\}$ (c)

$[x;y]\in\{[-1;1],[1;-1];[1;2];[2;1]\}$ (d)

$[x;y]\in\{[1;3];[3;1]\}$
(e)

$[x;y]\in\{[-1;2];[2;-1]\}$ (f)

$[x;y]\in\{[-\frac{1+\sqrt{33}}2;\frac{\sqrt{33}-1}2];[\frac{\sqrt{33}-1}2;-\frac{1+\sqrt{33}}2]\}$ (g)

$[x;y]\in\{[\frac12;\frac13]\}$
(h)

$[x;y]\in\{[3-\sqrt3;3+\sqrt3];[3+\sqrt3;3-\sqrt3]\}$ (i)

$[x;y]\in\{[0;2];[2;0]\}$ (j)

$[x;y]\in\{[5;3]\}$
(k)

$[x;y]\in\{[-\frac95;-\frac{16}5];[\frac95;\frac{16}5]\}$ (l)

$[x;y]\in\{[3;5];[5;3]\}$ (m)

$[x;y]\in\{[-1;-9];[4\pm2\sqrt2;8]\}$
(n)

$[x;y]\in\{[-4;1];[1;-4];[1;2];[2;1]\}$ (o)

$[x;y]\in\{[-5;-2];[-3;-4];[1;4];[3;2]\}$
(p)

$[x;y]\in\{[-3;-3];[2;2];[\frac{1-\sqrt{21}}2;\frac{1+\sqrt{21}}2];[\frac{1+\sqrt{21}}2;\frac{1-\sqrt{21}}2]\}$

Příklad 9

V množině $\mathbb R^2$ řešte soustavy

(a) $\begin{cases}x^2+3y^2=7\\ xy+y^2=3\end{cases}$	(b) $\begin{cases}x^2-4y^2=9\\ xy+2y^2=18\end{cases}$
(c) $\begin{cases}x^2+xy=15\\ y^2+xy=10\end{cases}$	(d) $\begin{cases}x^2-xy=28\\ y^2-xy=-12\end{cases}$
(e) $\begin{cases}x^2+3xy=54\\4y^2+xy=115\end{cases}$	(f) $\begin{cases}x^2+2y^2=17\\ x^2-2xy=-3\end{cases}$
(g) $\begin{cases} 3x^2+8y^2=140\\ 5x^2+8xy=84\end{cases}$	(h) $\begin{cases} x^2+4xy=0\\x^2-xy+y^2=21\end{cases}$
(i) $\begin{cases} x^2-xy+y^2=28\\2x^2+3xy-2y^2=0\end{cases}$	(j) $\begin{cases} x^2-xy-12y^2=0\\ x^2+xy-10y^2=20\end{cases}$
(k) $\begin{cases}x^2-3xy+2y^2=0\\2x^2+3xy-y^2=13 \end{cases}$	(l) $\begin{cases}2x^2-3xy+y^2=3\\ x^2+2xy-2y^2=6\end{cases}$
(m) $\begin{cases}x^2 - 3xy + y^2 = -5\\ 3x^2 - 8xy + 3y^2 = -9\end{cases}$	(n) $\begin{cases} x^2-3xy+2y^2=15\\ 2x^2+y^2=6\end{cases}$
(o) $\begin{cases} x^2-xy-12y^2=8\\x^2+xy-10y^2=20\end{cases}$	(p) $\begin{cases} 6x^2+3xy+2y^2=24\\ 3x^2+2xy+2y^2=18\end{cases}$

Řešení	Ukázat
(a) $[x;y]\in\{[-2;-1];[-\frac12;-\frac32];[\frac12;\frac32];[2;1]\}$ (b) $[x;y]\in\{[-5;-2];[5;2]\}$ (c) $[x;y]\in\{[-3;-2];[3;2]\}$ (d) $[x;y]\in\{[-7;-3];[7;3]\}$ (e) $[x;y]\in\{[-36;\frac{23}2];[-3;-5];[2;5];[36;-\frac{23}2]\}$ (f) $[x;y]\in\{[-3;-2];[3;2];[-\frac{\sqrt3}3;-\frac{5\sqrt3}3][\frac{\sqrt3}3;\frac{5\sqrt3}3];\}$ (g) $[x;y]\in\{[-6;2];[-2;-4];[2;4];[6;-2]\}$ (h) $[x;y]\in\{[-4;1];[0;\pm\sqrt{21}];[4;-1]\}$ (i) $[x;y]\in\{[-4;2];[4;-2];[-2\sqrt{\frac73};-4\sqrt{\frac73}];[2\sqrt{\frac73};4\sqrt{\frac73}]\}$ (j) $[x;y]\in\{[-4\sqrt2;-\sqrt2];[4\sqrt2;\sqrt2]\}$ (k) $[x;y]\in\{[-2;-1];[2;1];[-\frac{\sqrt{13}}2;-\frac{\sqrt{13}}2];[\frac{\sqrt{13}}2;\frac{\sqrt{13}}2]\}$ (l) $[x;y]\in\{[-2;-1];[2;1]\}$ (m) $[x;y]\in\{[-3;-2];[-2;-3];[2;3];[3;2]\}$ (n) $[x;y]\in\{[-1;2];[1;-2];[-\frac1{\sqrt3};\frac4{\sqrt3}];[\frac1{\sqrt3};-\frac4{\sqrt3}]\}$ (o) $[x;y]\in\{[-5;-1];[5;1]\}$ (p) $[x;y]\in\{[-2;3];[2;-3];[-\sqrt{\frac65};-2\sqrt{\frac65}];[\sqrt{\frac65};2\sqrt{\frac65}]\}$

Řešení

Ukázat

(a)

$[x;y]\in\{[-2;-1];[-\frac12;-\frac32];[\frac12;\frac32];[2;1]\}$ (b)

$[x;y]\in\{[-5;-2];[5;2]\}$ (c)

$[x;y]\in\{[-3;-2];[3;2]\}$
(d)

$[x;y]\in\{[-7;-3];[7;3]\}$ (e)

$[x;y]\in\{[-36;\frac{23}2];[-3;-5];[2;5];[36;-\frac{23}2]\}$
(f)

$[x;y]\in\{[-3;-2];[3;2];[-\frac{\sqrt3}3;-\frac{5\sqrt3}3][\frac{\sqrt3}3;\frac{5\sqrt3}3];\}$ (g)

$[x;y]\in\{[-6;2];[-2;-4];[2;4];[6;-2]\}$
(h)

$[x;y]\in\{[-4;1];[0;\pm\sqrt{21}];[4;-1]\}$ (i)

$[x;y]\in\{[-4;2];[4;-2];[-2\sqrt{\frac73};-4\sqrt{\frac73}];[2\sqrt{\frac73};4\sqrt{\frac73}]\}$
(j)

$[x;y]\in\{[-4\sqrt2;-\sqrt2];[4\sqrt2;\sqrt2]\}$ (k)

$[x;y]\in\{[-2;-1];[2;1];[-\frac{\sqrt{13}}2;-\frac{\sqrt{13}}2];[\frac{\sqrt{13}}2;\frac{\sqrt{13}}2]\}$ (l)

$[x;y]\in\{[-2;-1];[2;1]\}$ (m)

$[x;y]\in\{[-3;-2];[-2;-3];[2;3];[3;2]\}$ (n)

$[x;y]\in\{[-1;2];[1;-2];[-\frac1{\sqrt3};\frac4{\sqrt3}];[\frac1{\sqrt3};-\frac4{\sqrt3}]\}$
(o)

$[x;y]\in\{[-5;-1];[5;1]\}$ (p)

$[x;y]\in\{[-2;3];[2;-3];[-\sqrt{\frac65};-2\sqrt{\frac65}];[\sqrt{\frac65};2\sqrt{\frac65}]\}$

Příklad 10

V množině $\mathbb R^2$ řešte soustavy

(a) $\begin{cases}2x+5y+2xy=-19\\ x+y+3xy=-35\end{cases}$	(b) $\begin{cases} 240=xy\\ 240=(x+8)(y-1)\end{cases}$
(c) $\begin{cases}x+y+\dfrac xy=9\\ \dfrac{(x+y)x}y=20\end{cases}$	(d) $\begin{cases}\dfrac xy+\dfrac yx=\dfrac{13}6\\ x+y=5\end{cases}$
(e) $\begin{cases}(x+y+2)(x+y-2)=5\\x^2+xy+y^2=9\end{cases}$	(f) $\begin{cases}(2x-5)^2+(3y-2)^2=17\\ (2x-5)(3y-2)=4\end{cases}$
(g) $\begin{cases}\dfrac{x+y}{x-y}+\dfrac{x-y}{x+y}=\dfrac{13}6\\ xy=5\end{cases}$	(h) $\begin{cases}\dfrac xy-\dfrac yx=\dfrac56\\ x^2-y^2=5\end{cases}$
(i) $\begin{cases}\dfrac{x^2+y^2}{x+y}=\dfrac{10}3\\ \dfrac1x+\dfrac1y=\dfrac34\end{cases}$	(j) $\begin{cases}\dfrac5{x^2+xy}+\dfrac4{y^2+xy}=\dfrac{13}6\\ \dfrac8{x^2+xy}-\dfrac1{y^2+xy}=1\end{cases}$
(k) $\begin{cases}\dfrac xy+\dfrac yx=\dfrac52\\ x^2-y^2=3\end{cases}$	(l) $\begin{cases}\dfrac xy+\dfrac yx=\dfrac{10}3\\ x^2-y^2=8\end{cases}$
(m) $\begin{cases} (x^2+y^2)(x-y)=447\\ (x-y)xy=210 \end{cases}$	(n) $\begin{cases}\dfrac{x+2y}{x-y}+\dfrac{x-2y}{x+y}=4\\ x^2+xy+y^2=21\end{cases}$
(o) $\begin{cases}x^{-1}+y^{-1}=0\\ x^{-2}+y^{-2}=8\end{cases}$	(p) $\begin{cases}x^{-1}+y^{-1}=5\\ x^{-2}+y^{-2}=13\end{cases}$
(q) $\begin{cases}x+y+\dfrac xy=9\\ \dfrac{(x+y)x}y=20\end{cases}$	(r) $\begin{cases}xy-\dfrac xy=\dfrac{16}3\\ xy-\dfrac yx=\dfrac92\end{cases}$

Řešení	Ukázat
(a) $[x;y]\in\{[2\pm\sqrt{43};\frac{5\mp4\sqrt{43}}{13}]\}$ (b) $[x;y]\in\{[-48;-5];[40;6]\}$ (c) $[x;y]\in\{[\frac{10}3;\frac23];[4;1]\}$ (d) $[x;y]\in\{[2;3];[3;2]\}$ (e) $[x;y]\in\{[\pm3;0];[0;\pm3]\}$ (f) $[x;y]\in\{[\frac12;\frac13];[2;-\frac23];[3;2];[\frac92;1]\}$ (g) $[x;y]\in\{[-5;-1];[5;1]\}$ (h) $[x;y]\in\{[-3;-2];[3;2]\}$ (i) $[x;y]\in\{[2;4];[4;2]\}$ (j) $[x;y]\in\{[-2;-1];[2;1]\}$ (k) $[x;y]\in\{[-2;-1];[2;1]\}$ (l) $[x;y]\in\{[3;1];[-3;-1]\}$ (m) $[x;y]\in\{[-7;-10];[10;7]\}$ (n) $[x;y]\in\{[-2\sqrt3;-\sqrt3];[2\sqrt3;\sqrt3];[-2\sqrt7;\sqrt7];[2\sqrt7;\sqrt7]\}$ (o) $[x;y]\in\{[-\frac12;\frac12];[\frac12;-\frac12]\}$ (p) $[x;y]\in\{[\frac12;\frac3];[\frac13;\frac12]\}$ (q) $[x;y]\in\{[\frac{10}3;\frac23];[4;1]\}$ (r) $[x;y]\in\{[-2;-3];[2;3]\}$

Řešení

Ukázat

(a)

$[x;y]\in\{[2\pm\sqrt{43};\frac{5\mp4\sqrt{43}}{13}]\}$ (b)

$[x;y]\in\{[-48;-5];[40;6]\}$ (c)

$[x;y]\in\{[\frac{10}3;\frac23];[4;1]\}$ (d)

$[x;y]\in\{[2;3];[3;2]\}$
(e)

$[x;y]\in\{[\pm3;0];[0;\pm3]\}$ (f)

$[x;y]\in\{[\frac12;\frac13];[2;-\frac23];[3;2];[\frac92;1]\}$ (g)

$[x;y]\in\{[-5;-1];[5;1]\}$
(h)

$[x;y]\in\{[-3;-2];[3;2]\}$ (i)

$[x;y]\in\{[2;4];[4;2]\}$ (j)

$[x;y]\in\{[-2;-1];[2;1]\}$ (k)

$[x;y]\in\{[-2;-1];[2;1]\}$
(l)

$[x;y]\in\{[3;1];[-3;-1]\}$ (m)

$[x;y]\in\{[-7;-10];[10;7]\}$
(n)

$[x;y]\in\{[-2\sqrt3;-\sqrt3];[2\sqrt3;\sqrt3];[-2\sqrt7;\sqrt7];[2\sqrt7;\sqrt7]\}$ (o)

$[x;y]\in\{[-\frac12;\frac12];[\frac12;-\frac12]\}$ (p)

$[x;y]\in\{[\frac12;\frac3];[\frac13;\frac12]\}$
(q)

$[x;y]\in\{[\frac{10}3;\frac23];[4;1]\}$ (r)

$[x;y]\in\{[-2;-3];[2;3]\}$

Příklad 11

V množině $\mathbb R^2$ řešte soustavy

(a) $\begin{cases}\dfrac4{x+y}+\dfrac4{x-y}=3\\ (x+y)^2+(x-y)^2=20\end{cases}$	(b) $\begin{cases}\dfrac4{x+y-1}-\dfrac5{2x-y+3}+\dfrac52=0\\ \dfrac3{x+y-1}+\dfrac1{2x-y+3}+\dfrac75=0\end{cases}$
(c) $\begin{cases}\dfrac5{x^2-xy}+\dfrac4{y^2-xy}=-\dfrac16\\ \dfrac7{x^2-xy}-\dfrac3{y^2-xy}=\dfrac65\end{cases}$	(d) $\begin{cases}\dfrac3{x^2+y^2-1}+\dfrac{2y}{x}=1\\ x^2+y^2+\dfrac{4x}y=22\end{cases}$
(e) $\begin{cases}x+y+\dfrac1{x-y}=\dfrac{16}5\\ x-y+\dfrac1{x+y}=\dfrac{16}3\end{cases}$	(f) $\begin{cases}x+y+\dfrac{x^2}{y^2}=7\\ \dfrac{(x+y)x^2}{y^2}=12\end{cases}$

Řešení	Ukázat
(a) $x\in\{[3;\pm1];[-\frac53;\pm\frac{\sqrt{65}}3]\}$ (b) $x\in\{[2;-3]\}$ (c) $x\in\{[-5;-3];[5;3]\}$ (d) $x\in\{[-3;-1];[3;1];[-14\sqrt{\frac2{53}};-4\sqrt{\frac2{53}}];[14\sqrt{\frac2{53}};4\sqrt{\frac2{53}}]\}$ (e) $x\in\{[4;1];[\frac4{15};-\frac1{15}]\}$ (f) $x\in\{[2;1];[6;-3];[6-2\sqrt3;2\sqrt3-2];[6+2\sqrt3;-2-2\sqrt3]\}$

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