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) $\begin{cases}x-3y=10\\ x^2-24y=100\end{cases}$
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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) $\begin{cases}x-\dfrac{x-y}2=4\\ y-\dfrac{x+3y}{x+2}=1\end{cases}$
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Příklad 3

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

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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

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) $\begin{cases}\dfrac1{y-1}-\dfrac1{y+1}=\dfrac1{x^2}\\ x^2-y-5=0\end{cases}$
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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) $\begin{cases}x^2+y^2-2x+3y=9\\ 2x^2+2y^2-3x+5y=17\end{cases}$
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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

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) $\begin{cases}x^2+y=y^2+x\\ y^2+x=6\end{cases}$
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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) $\begin{cases} 6x^2+3xy+2y^2=24\\ 3x^2+2xy+2y^2=18\end{cases}$
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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) $\begin{cases}xy-\dfrac xy=\dfrac{16}3\\ xy-\dfrac yx=\dfrac92\end{cases}$

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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) $\begin{cases}x+y+\dfrac{x^2}{y^2}=7\\ \dfrac{(x+y)x^2}{y^2}=12\end{cases}$
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