Příklad 1

V množině $\mathbb R^2$ řešte soustavy

(a) $\begin{cases}x^2y^3=\frac12\\ x^3y^2=2\end{cases}$	(b) $\begin{cases}x^2y^3=8\\ x^3y^2=4\end{cases}$
(c) $\begin{cases}x^2y^3=16\\ x^3y^2=2\end{cases}$	(d) $\begin{cases}x^2y^3=16\\ x^3y^2=64\end{cases}$
(e) $\begin{cases}x^2y^3=108\\ x^3y^2=72\end{cases}$	(f) $\begin{cases}x^2y^3+x^3y^2=12\\ x^2y^3-x^3y^2=4\end{cases}$

Řešení	Ukázat
(a) $[x;y]=[2;\frac12]$ (b) $[x;y]=[1;2]$ (c) $[x;y]=[\frac12;4]$ (d) $[x;y]=[4;1]$ (e) $[x;y]=[2;3]$ () $[x;y]=[1;2]$

Příklad 2

V množině $\mathbb R^2$ řešte soustavy

(a) $\begin{cases}x+y^2=7\\ xy^2=12\end{cases}$	(b) $\begin{cases} x-y=1\\x^3-y^3=7 \end{cases}$
(c) $\begin{cases} x^3-y^3=28\\ x-y=4\end{cases}$	(d) $\begin{cases} x^3+y^3=35\\x+y=5 \end{cases}$
(e) $\begin{cases}x^3+y^3=9\\ x+y=3\end{cases}$	(f) $\begin{cases}(x-y)(x^2-y^2)=45\\ x+y=5\end{cases}$
(g) $\begin{cases}\dfrac{x^2}y+\dfrac{y^2}x=18\\ x+y=12\end{cases}$	(h) $\begin{cases} x^3+y^3=9\\xy=2 \end{cases}$

Řešení	Ukázat
(a) $[x;y]\in\{[3;\pm2];[4\pm\sqrt3;]\}$ (b) $[x;y]\in\{[-1;2];[2;1]\}$ (c) $[x;y]\in\{[1;-3];[3;-1]\}$ (d) $[x;y]\in\{[2;3];[3;2]\}$ (e) $[x;y]\in\{[1;2];[2;1]\}$ (f) $[x;y]\in\{[1;4];[4;1]\}$ (g) $[x;y]\in\{[4;8];[8;4]\}$ (h) $[x;y]\in\{[1;2];[2;1]\}$

Příklad 3

V množině $\mathbb R^2$ řešte soustavy

(a) $\begin{cases} x^3-y^3=26\\x^2y-xy^2=6 \end{cases}$	(b) $\begin{cases} x^{3}+y^{3}=65\\x^{2}y+y^{2}x=20\end{cases}$
(c) $\begin{cases}x^3+y^3=7\\xy(x+y)=-2\end{cases}$	(d) $\begin{cases}xy(x+y)=30\\ x^3+y^3=35\end{cases}$
(e) $\begin{cases}x^3+y^3=19\\ x^2y+xy^2=-6\end{cases}$	(f) $\begin{cases}x^3+y^3=1\\x^2y+xy^2=1\end{cases}$
(g) $\begin{cases}xy+x-y=3\\x^2y-xy^2=2\end{cases}$	(h) $\begin{cases}x+xy+y=11\\ x^2y+xy^2=30\end{cases}$
(i) $\begin{cases}x^2y+xy^2=6\\ xy+x+y=5\end{cases}$	(j) $\begin{cases} x^3-y^3=19\\ x^2+xy+y^2=19\end{cases}$
(k) $\begin{cases}x^2y+xy^2=30\\ \dfrac1x+\dfrac1y=\dfrac56\end{cases}$	(l) $\begin{cases}\dfrac{x^2}y+\dfrac{y^2}x=3\\ x+y=2\end{cases}$

Řešení	Ukázat
(a) $[x;y]\in\{[-1;-3];[3;1]\}$ (b) $[x;y]\in\{[1;4];[4;1]\}$ (c) $[x;y]\in\{[-1;2];[2;-1]\}$ (d) $[x;y]\in\{[2;3];[3;2]\}$ (e) $[x;y]\in\{[-2;3];[3;-2]\}$ (f) $[x;y]\in\{[\frac1{\sqrt[3]2};\frac1{\sqrt[3]2}]\}$ (g) $[x;y]\in\{[-1;-2];[2;1];[1\pm\sqrt2;-1\pm\sqrt2]\}$ (h) $[x;y]\in\{[1;5];[2;3];[3;2];[5;1]\}$ (i) $[x;y]\in\{[1;2];[2;1]\}$ (j) $[x;y]\in\{[-2;-3];[3;2]\}$ (k) $[x;y]\in\{[-6;1];[1;-6];[2;3];[3;2]\}$ (l) $[x;y]\in\{[\frac23;\frac43];[\frac43;\frac23]\}$

Řešení

Ukázat

(a)

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

$[x;y]\in\{[1;4];[4;1]\}$ (c)

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

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

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

$[x;y]\in\{[\frac1{\sqrt[3]2};\frac1{\sqrt[3]2}]\}$ (g)

$[x;y]\in\{[-1;-2];[2;1];[1\pm\sqrt2;-1\pm\sqrt2]\}$
(h)

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

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

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

$[x;y]\in\{[-6;1];[1;-6];[2;3];[3;2]\}$ (l)

$[x;y]\in\{[\frac23;\frac43];[\frac43;\frac23]\}$

Příklad 4

V množině $\mathbb R^2$ řešte soustavy

(a) $\begin{cases}x^3+y^3=7x+7y\\ x^2+y^2=10\end{cases}$	(b) $\begin{cases}(x^2-3xy+2y^2)(x-y)=0\\x^2+y^2=10\end{cases}$
(c) $\begin{cases}x^3-y^3=26\\ x^2-y^2=2(x+y)\end{cases}$	(d) $\begin{cases}x^3-y^3=19(x-y)\\ x^3+y^3=7(x+y)\end{cases}$
(e) $\begin{cases}x^3+y^3=7\\ x^3y^3=-8\end{cases}$	(f) $\begin{cases}x^3-y^3-3x^2+3y^2=-2\\x^2-y^2=1\end{cases}$
(g) $\begin{cases}x^3+3xy^2=158\\ 3x^2y+y^3=-185 \end{cases}$	(h) $\begin{cases}x^3+xy^2=40y\\ y^3+x^2y=10x\end{cases}$
(i) $\begin{cases}(x^2+1)(y^2+1)=10\\ (x+y)(xy-1)=3\end{cases}$	(j) $\begin{cases}(x+y)(xy-1)=25\\ (x+y)(xy+1)=35\end{cases}$
(k) $\begin{cases}(x+y)(x^2-y^2)=16\\ (x-y)(x^2+y^2)=40 \end{cases}$	(l) $\begin{cases}(x-y)(x^2+y^2)=5\\ (x+y)(x^2-y^2)=9\end{cases}$

Řešení	Ukázat
(a) $[x;y]\in\{[\pm3;\pm1];[\pm1;\pm3];[\pm\sqrt5;\mp\sqrt5\}$ (b) $[x;y]\in\{[\pm2\sqrt2;\pm\sqrt2];\pm\sqrt5;\pm\sqrt5\}$ (c) $[x;y]\in\{[-1;-3];[3;1];[\sqrt[3]{13};-\sqrt[3]{13}\}$ (d) $[x;y]\in\{[\pm3;\pm2];[0;0];[\pm2;\pm3];[\pm\sqrt7;\pm\sqrt7];[\pm\sqrt{19};\mp\sqrt{19}]\}$ (e) $[x;y]\in\{[-1;2];[2;-1]\}$ (f) $[x;y]\in\{[1;0]\}$ (g) $[x;y]\in\{[2;-5]\}$ (h) $[x;y]\in\{[\pm4;\pm2];[0;0]\}$ (i) $[x;y]\in\{[-3;0];[\pm2;1];[1;\pm2];[0;-3]\}$ (j) $[x;y]\in\{[2;3];[3;2]\}$ (k) $[x;y]\in\{[1;-3];[3;-1]\}$ (l) $[x;y]\in\{[-1;-2];[2;1]\}$

Řešení

Ukázat

(a)

$[x;y]\in\{[\pm3;\pm1];[\pm1;\pm3];[\pm\sqrt5;\mp\sqrt5\}$ (b)

$[x;y]\in\{[\pm2\sqrt2;\pm\sqrt2];\pm\sqrt5;\pm\sqrt5\}$
(c)

$[x;y]\in\{[-1;-3];[3;1];[\sqrt[3]{13};-\sqrt[3]{13}\}$ (d)

$[x;y]\in\{[\pm3;\pm2];[0;0];[\pm2;\pm3];[\pm\sqrt7;\pm\sqrt7];[\pm\sqrt{19};\mp\sqrt{19}]\}$
(e)

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

$[x;y]\in\{[1;0]\}$ (g)

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

$[x;y]\in\{[\pm4;\pm2];[0;0]\}$
(i)

$[x;y]\in\{[-3;0];[\pm2;1];[1;\pm2];[0;-3]\}$ (j)

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

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

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

Příklad 5

V množině $\mathbb R^2$ řešte soustavy

(a) $\begin{cases} x+y=1\\x^4+y^4=7 \end{cases}$	(b) $\begin{cases} x+y=3\\x^4+y^4=17 \end{cases}$
(c) $\begin{cases} x^4-y^4=45\\x^2-y^2=5 \end{cases}$	(d) $\begin{cases} x^4+y^4=82\\xy=3 \end{cases}$
(e) $\begin{cases} x^4+y^4=17\\x^2+y^2=5 \end{cases}$	(f) $\begin{cases} x^2+y^4=20\\x^4+y^2=20 \end{cases}$
(g) $\begin{cases} x^2+y^4=5\\xy^2=2\end{cases}$	(h) $\begin{cases}x^4+y^4=x^2y^2+13\\x^2+y^2=xy+3\end{cases}$
(i) $\begin{cases}x^4+x^2y^2+y^4=91\\ x^2+xy+y^2=13\end{cases}$	(j) $\begin{cases}x^6+y^6=65\\ x^4-x^2y^2+y^4=13\end{cases}$
(k) $\begin{cases}x^4-y^4=15\\ x^3y-xy^3=6\end{cases}$	(l) $\begin{cases}x^4+y^4+x^2+y^2=92\\ xy=3\end{cases}$
(m) $\begin{cases}x^3+x^3y^3+y^3=12\\ x+xy+y=0\end{cases}$	(n) $\begin{cases}x^2+y^4=5\\ xy^2=2\end{cases}$

Řešení	Ukázat
(a) nápověda: Pár perliček, př. 8 $[x;y]\in\{[\frac{1\pm\sqrt5}2;\frac{1\mp\sqrt5}2]\}$ (b) $[x;y]\in\{[1;2];[2;1]\}$ (c) $[x;y]\in\{[-\sqrt7;\pm\sqrt2];[\sqrt7;\pm\sqrt2]\}$ (d) $[x;y]\in\{[\pm3;\pm1];[\pm1;\pm3]\}$ (e) nápověda $x^2+y^2=a$ , $x^2y^2=b$ $[x;y]\in\{[-2;\pm1];[-1;\pm2];[1;\pm2];[2;\pm-1]\}$ (f) $[x;y]\in\{[-2;\pm2];[2;\pm2]\}$ (g) $[x;y]\in\{[1;\pm\sqrt2];[2;\pm1]\}$ (h) nápověda: $x^2+y^2=a$ , $xy=b$ $[x;y]\in\{[\pm2;\pm1];[\pm1;\pm2];[\pm\sqrt{2-\sqrt3};\pm\sqrt{2-\sqrt3}(2+\sqrt3)];[\pm\sqrt{2+\sqrt3};\mp\sqrt{2+\sqrt3}(\sqrt3-2)]\}$ (i) $[x;y]\in\{[\pm3;\pm1];[\pm1;\pm3]\}$ (j) nápověda: $x^2+y^2=a$ , $x^2y^2=b$ $[x;y]\in\{[-2;\pm1];[-1;\pm2];[1;\pm2];[2;\pm1]\}$ (k) nápověda: $y=kx$ $[x;y]\in\{[-2;-1];[2;1]\}$ (l) $[x;y]\in\{[\pm3;\pm1];[\pm1;\pm3\}$ (m) $[x;y]\in\{[1-\sqrt3;1+\sqrt3];[1+\sqrt3;1-\sqrt3]\}$ (n) $[x;y]\in\{[1;\pm\sqrt2];[2;\pm1]\}$

Řešení

Ukázat

(a) nápověda: Pár perliček, př. 8

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

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

$[x;y]\in\{[-\sqrt7;\pm\sqrt2];[\sqrt7;\pm\sqrt2]\}$ (d)

$[x;y]\in\{[\pm3;\pm1];[\pm1;\pm3]\}$ (e) nápověda

$x^2+y^2=a$ ,

$x^2y^2=b$

$[x;y]\in\{[-2;\pm1];[-1;\pm2];[1;\pm2];[2;\pm-1]\}$
(f)

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

$[x;y]\in\{[1;\pm\sqrt2];[2;\pm1]\}$ (h) nápověda:

$x^2+y^2=a$ ,

$xy=b$

$[x;y]\in\{[\pm2;\pm1];[\pm1;\pm2];[\pm\sqrt{2-\sqrt3};\pm\sqrt{2-\sqrt3}(2+\sqrt3)];[\pm\sqrt{2+\sqrt3};\mp\sqrt{2+\sqrt3}(\sqrt3-2)]\}$
(i)

$[x;y]\in\{[\pm3;\pm1];[\pm1;\pm3]\}$ (j) nápověda:

$x^2+y^2=a$ ,

$x^2y^2=b$

$[x;y]\in\{[-2;\pm1];[-1;\pm2];[1;\pm2];[2;\pm1]\}$
(k) nápověda:

$y=kx$

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

$[x;y]\in\{[\pm3;\pm1];[\pm1;\pm3\}$
(m)

$[x;y]\in\{[1-\sqrt3;1+\sqrt3];[1+\sqrt3;1-\sqrt3]\}$ (n)

$[x;y]\in\{[1;\pm\sqrt2];[2;\pm1]\}$

Příklad 6

V množině $\mathbb R^2$ řešte soustavy

(a) $\begin{cases} \sqrt[3]{x}+\sqrt[3]{y}=3\\xy=8 \end{cases}$	(b) $\begin{cases} \sqrt[3]{x}+\sqrt[3]{y}=4\\xy=27 \end{cases}$
(c) $\begin{cases} \sqrt[3]{x}+\sqrt[3]{y}=4\\x+y=28 \end{cases}$	(d) $\begin{cases} x\sqrt{y}+y\sqrt{x}=6\\x^2y+y^2x=20 \end{cases}$
(e) $\begin{cases} x-\sqrt{xy}=28\\ y-\sqrt{xy}=-12 \end{cases}$	(f) $\begin{cases} \sqrt{xy+1}=2\\y-x=2 \end{cases}$
(g) $\begin{cases} x\sqrt y+y\sqrt x=30\\ x\sqrt x+y\sqrt y=35\end{cases}$	(h) $\begin{cases}\sqrt{x+y+2}=2\\ x^2+y^2=4 \end{cases}$
(i) $\begin{cases}\sqrt{\dfrac{x+y}2}+\sqrt{\dfrac{x-y}3}=14\\\sqrt{\dfrac{x+y}8}-\sqrt{\dfrac{x-y}{12}}=3 \end{cases}$	(j) $\begin{cases}\dfrac1{\sqrt x}+\dfrac1{\sqrt y}=\dfrac43\\ xy=9 \end{cases}$
(k) $\begin{cases}\sqrt{(x+y)^2}=3\\\sqrt{(x-y)^2}=1 \end{cases}$	(l) $\begin{cases}\sqrt x+\sqrt y=10\\\sqrt[4]x+\sqrt[4]y=4 \end{cases}$
(m) $\begin{cases}\sqrt[4]x-\sqrt[4]y=1\\ \sqrt x+\sqrt y=5 \end{cases}$	(n) $\begin{cases}\sqrt[4]{x+y}+\sqrt[4]{x-y}=4\\ \sqrt{x+y}-\sqrt{x-y}=8 \end{cases}$
(o) $\begin{cases}\sqrt[3]{\dfrac{x+y}{x-y}}-\sqrt[3]{\dfrac{x-y}{x+y}}=\dfrac32\\ x^2-y^2=32\end{cases}$	(p) $\begin{cases}\sqrt{\dfrac xy+2+\dfrac yx}=\dfrac52\\\|x+y\|=5\end{cases}$
(q) $\begin{cases}\sqrt{\dfrac{2x-1}{y+2}}+\sqrt{\dfrac{y+2}{2x-1}}=2\\ x+y=12\end{cases}$	(r) $\begin{cases}x-y+\sqrt{x^2-4y^2}=2\\ x^5\cdot\sqrt{x^2-4y^2}=0\end{cases}$

Řešení	Ukázat
(a) $[x;y]\in\{[1;8];[8;1]\}$ , (b) $[x;y]\in\{[1;27];[27;1]\}$ (c) $[x;y]\in\{[1;27];[27;1]\}$ (d) $[x;y]\in\{[1;4];[4;1]\}$ (e) $[x;y]\in\{[49;9]\}$ (f) $[x;y]\in\{[-3;-1];[1;3]\}$ (g) $[x;y]\in\{[4;9];[9;4]\}$ (h) $[x;y]\in\{[0;2];[2;0]\}$ (i) $[x;y]\in\{[124;76]\}$ (j) $[x;y]\in\{[1;9];[9;1]\}$ (k) $[x;y]\in\{[-2;-1];[-1;-2];[1;2];[2;1]\}$ (l) $[x;y]\in\{[1;81];[81;1]\}$ (m) $[x;y]\in\{[16;1]\}$ (n) $[x;y]\in\{[41;40]\}$ (p) $[x;y]\in\{[-9;-7];[9;7]\}$ (p) $[x;y]\in\{[\pm1;\pm4];[\pm4;\pm1]\}$ (q) $[x;y]\in\{[5;7]\}$ (r) $[x;y]\in\{[4;2];[\frac43;-\frac23]\}$

Řešení

Ukázat

(a)

$[x;y]\in\{[1;8];[8;1]\}$ , (b)

$[x;y]\in\{[1;27];[27;1]\}$ (c)

$[x;y]\in\{[1;27];[27;1]\}$ (d)

$[x;y]\in\{[1;4];[4;1]\}$ (e)

$[x;y]\in\{[49;9]\}$ (f)

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

$[x;y]\in\{[4;9];[9;4]\}$ (h)

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

$[x;y]\in\{[124;76]\}$ (j)

$[x;y]\in\{[1;9];[9;1]\}$ (k)

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

$[x;y]\in\{[1;81];[81;1]\}$ (m)

$[x;y]\in\{[16;1]\}$ (n)

$[x;y]\in\{[41;40]\}$ (p)

$[x;y]\in\{[-9;-7];[9;7]\}$ (p)

$[x;y]\in\{[\pm1;\pm4];[\pm4;\pm1]\}$ (q)

$[x;y]\in\{[5;7]\}$ (r)

$[x;y]\in\{[4;2];[\frac43;-\frac23]\}$

Příklad 7

V množině $\mathbb R^3$ řešte soustavy

(a)

$\begin{cases}x^2+y^2=13^2\\x^2+z^2=15^2\\y+z=14\end{cases}$

(b)

$\begin{cases}x^2-y=z^2\\y^2-z=x^2\\z^2-x=y^2\end{cases}$

(c)

$\begin{cases}x+yz=2\\y+xz=2\\z+xy=2\end{cases}$

Řešení	Ukázat
(a) $[x;y;z]\in\{[\pm12;5;9]\}$ (b) $[x;y;z]\in\{[0;0;0];[0;-1;1];[1;0;-1];[-1;1;0]\}$ (c) $[x;y;z]\in\{[1;1;1];[-2;-2;-2]\}$

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