Příklad 1

V $\mathbb{R}^2$ rešte soustavy rovnic:

(a) $\begin{cases}x+y=5\\ x-y=1\end{cases}$	(b) $\begin{cases}10x-8y=-5x\\ 5x-4y=-20\end{cases}$
(c) $\displaystyle \left\{\begin{array}{l} x+2y=11\\ 3x+y=13 \end{array}\right.$	(d) $\displaystyle \left\{\begin{array}{l} 4x+3y=6\\ 2x+y=4 \end{array}\right.$
(e) $\displaystyle \left\{\begin{array}{l} 4x+3y=-4\\ 6x+5y=-7 \end{array}\right.$	(f) $\displaystyle \left\{\begin{array}{l} x+15y=53\\ 3x+y=27 \end{array}\right.$
(g) $\displaystyle \left\{\begin{array}{l} 3x-5y=14\\ 6x-10y=17 \end{array}\right.$	(h) $\displaystyle \left\{\begin{array}{l} 7x+3y=100\\ 14x+6y=200 \end{array}\right.$
(i) $\displaystyle \left\{\begin{array}{l} 3x-5y=11\\ 6x-10y=22 \end{array}\right.$	(j) $\displaystyle \left\{\begin{array}{l} x=20-3y\\ x=5y+12 \end{array}\right.$
(k) $\displaystyle\left\{\begin{array}{l}3x + 2y = 11\\ 2x-3y = 10\end{array}\right.$	(l) $\displaystyle\left\{\begin{array}{l} x-y = 0\\ -2x + 2y = 3\end{array}\right.$
(m) $\displaystyle\begin{cases}x-2y=6\\ 5x+2y=6\end{cases}$	(n) $\displaystyle\begin{cases}7x-2y=5\\ x+y=12\end{cases}$
(o) $\displaystyle\begin{cases}2x-3y=1\\ x+4y=6\end{cases}$	(p) $\displaystyle\begin{cases}3x+2y=13\\ x+3y=2\end{cases}$
(q) $\displaystyle \left\{\begin{array}{l} 2(x+y)-5(y-x)=17\\ 3(x+2y)+7(3x+5y)=7 \end{array}\right.$	(r) $\begin{cases}2x+7y-18=4(x+y)\\ 5x-4y-13=2(x+y)\end{cases}$
(s) $\begin{cases}x-2y=6\\ 5x+2y=4\end{cases}$	(t) $\begin{cases}7x-2y=12\\ x+y=12\end{cases}$

Řešení	Ukázat
(a) $[x;y]=[3;2]$ (b) $[x;y]=[8;15]$ (c) $[x;y]=[3;4]$ (d) $[x;y]=[3;-2]$ (e) $[x;y]=[-\frac12;-2]$ (f) $[x;y]=[8;3]$ (g) nemá řešení (h) $[x;y]=[t;\frac{100-7t}3]\ t\in\mathbb R$ (i) $[x;y]=[t;\frac{3t-11}5]\ t\in\mathbb R$ (j) $[x;y]=[17;1]$ (k) $[x;y]=[\frac{53}{13};-\frac8{13}]$ (l) nemá řešení (m) $[x;y]=[2;-2]$ (n) $[x;y]=[\frac{29}9;\frac{79}9]$ (o) $[x;y]=[2;1]$ (p) $[x;y]=[5;-1]$ (q) $[x;y]=[2;-1]$ (r) $[x;y]=[-49;-\frac{80}3]$ (s) $[x;y]=[2;-2]$ (t) $[x;y]=[4;8]$

Řešení

Ukázat

(a)

$[x;y]=[3;2]$ (b)

$[x;y]=[8;15]$ (c)

$[x;y]=[3;4]$ (d)

$[x;y]=[3;-2]$ (e)

$[x;y]=[-\frac12;-2]$ (f)

$[x;y]=[8;3]$ (g) nemá řešení (h)

$[x;y]=[t;\frac{100-7t}3]\ t\in\mathbb R$ (i)

$[x;y]=[t;\frac{3t-11}5]\ t\in\mathbb R$ (j)

$[x;y]=[17;1]$ (k)

$[x;y]=[\frac{53}{13};-\frac8{13}]$ (l) nemá řešení (m)

$[x;y]=[2;-2]$ (n)

$[x;y]=[\frac{29}9;\frac{79}9]$ (o)

$[x;y]=[2;1]$ (p)

$[x;y]=[5;-1]$ (q)

$[x;y]=[2;-1]$ (r)

$[x;y]=[-49;-\frac{80}3]$ (s)

$[x;y]=[2;-2]$ (t)

$[x;y]=[4;8]$

Příklad 2

V $\mathbb{R}^2$ rešte soustavy rovnic:

(a) $\left\{\begin{array}{l} y=-\dfrac{1}{3}x+2\\ \dfrac{y}{2}+\dfrac{x}{6}=1\end{array}\right.$	(b) $\left\{\begin{array}{l} \dfrac{x+y}{5}+\dfrac{y}{5}=-2\\ \dfrac{2x-y}{3}-\dfrac{3x}{4}=\dfrac{3}{2} \end{array}\right.$
(c) $\left\{\begin{array}{l} \dfrac{4}{x-3y}=\dfrac{7}{9x+2y}\\ \dfrac{3}{2x+y}=\dfrac{9}{x-y+1} \end{array}\right.$	(d) $\left\{\begin{array}{l} \dfrac{2}{x-2y}=\dfrac{3}{2x-y}\\ \dfrac{4x-2y}{3(x-2y)}=1 \end{array}\right.$
(e) $\displaystyle \left\{\begin{array}{l} \dfrac{3x-4}{3y+4}=\dfrac12\\ \dfrac{2x-y}{2x+y}=\dfrac14 \end{array}\right.$	(f) $\begin{cases}(x+5)(y-2)=(x+2)(y-1)\\ (x-4)(y+7)=(x-3)(y+4)\end{cases}$
(g) $\displaystyle \left\{\begin{array}{l} (x+4)(y-2)=(x-5)(y+4)\\ (x+6)(y-1)=(x-1)(y+2) \end{array}\right.$	(h) $\begin{cases}(x+3)(y+5)=(x+1)(y+8)\\ (2x-3)(5y+7)=2(5x-6)(y+1)\end{cases}$

Řešení	Ukázat
(a) $[x;y]=[3t;2-t]$ $t\in\mathbb R$ (b) $[x;y]=[-2;-4]$ (c) $[x;y]=[1;-1]$ (d) $[x;y]=[t;-\frac t4]$ $t\in\mathbb R-\{0\}$ (e) $[x;y]=[5;6]$ (f) $[x;y]=[7;5]$ (g) $[x;y]=[8;4]$ (h) $[x;y]=[3;1]$

Příklad 3

V $\mathbb{R}^2$ rešte soustavy rovnic:

(a) $\displaystyle \left\{\begin{array}{l} 5(x-3)-3(y+2)=23 \\3(x-3)+5(y+2)=7\end{array}\right.$	(b) $\begin{cases}3(2x+3y)+2(2x-3y)=43\\ 8(2x+3y)-3(2x-3y)=73\end{cases}$
(c) $\displaystyle \left\{\begin{array}{l} \dfrac{1}{x}+\dfrac{3}{y}=5\\ \dfrac{2}{x}-\dfrac{6}{y}=6 \end{array}\right.$	(d) $\displaystyle \left\{\begin{array}{l} \dfrac{2}{x}+\dfrac{3}{y}=\dfrac{17}{2}\\\dfrac{1}{x}+\dfrac{2}{y}=5 \end{array}\right.$
(e) $\displaystyle \left\{\begin{array}{l} \dfrac{4}{x}+\frac{3}{y}=17\\\dfrac{1}{x}-\dfrac{3}{y}=-7 \end{array}\right.$	(f) $\displaystyle \begin{cases}\dfrac{2}{x-1}+\dfrac{1}{y+1}=1\\\dfrac{3}{x-1}+\dfrac{1}{y+1}=2\end{cases}$
(g) $\displaystyle \left\{\begin{array}{l}\dfrac{10}{x+5}+\dfrac{1}{y+2} = 1\\\dfrac{25}{x+5} + \dfrac{2}{y+2} = 1\end{array}\right.$	(h) $\displaystyle \left\{\begin{array}{l} \dfrac{1}{1-x+y}+\dfrac{1}{1-x-y}=\dfrac{2}{3}\\\dfrac{1}{1-x-y}-\dfrac{1}{1-x+y}=-\dfrac{4}{3} \end{array}\right.$
(i) $\displaystyle \left\{\begin{array}{l} \dfrac{2}{x+y}-\dfrac{5}{x-y}=1\\\dfrac{1}{x+y}+\dfrac{4}{x-y}=\dfrac{9}{5} \end{array}\right.$	(j) $\displaystyle \left\{\begin{array}{l} \dfrac{x+1}{x+y}+\dfrac{y}{x-y}=\dfrac{3}{2}\\\dfrac{x+1}{x+y}-3\cdot\dfrac{y}{x-y}=\dfrac{1}{2} \end{array}\right.$
(k) $\begin{cases}\dfrac{4}{x+y-1}-\dfrac{5}{2x-y+3}=-\dfrac{5}{2}\\ \dfrac{3}{x+y-1}+\dfrac{1}{2x-y+3}=-\dfrac{7}{5}\end{cases}$	(l) $\begin{cases}\dfrac2x+\dfrac y3=3\\ \dfrac x2+\dfrac3y=\dfrac32\end{cases}$

Řešení	Ukázat
(a) $[x;y]=[7;-3]$ (b) $[x;y]=[4;1]$ (c) $[x;y]=[\frac14;3]$ (d) $[x;y]=[\frac12;\frac23]$ (e) $[x;y]=[\frac12;\frac13]$ (f) $[x;y]=[2;-2]$ (g) $[x;y]=[-10;-\frac53]$ (h) $[x;y]=[2;2]$ (i) $[x;y]=[3;-2]$ (j) $[x;y]=[2;\frac25]$ (k) $[x;y]=[2;-3]$ (l) $[x;y[\in\{[1;3];[2;6]\}$

Příklad 4

V $\mathbb{R}^3$ rešte soustavy rovnic:

(a) $\left\{\begin{array}{l} x+y=7\\ x-2z=8\\ y+3z=5\end{array}\right.$	(b) $\left\{\begin{array}{l} x+y=28\\ y+z=32\\ x+z=30\end{array}\right.$
(c) $\left\{\begin{array}{l} x-y=\frac{1}{3}\\ y-z=\frac{1}{6}\\ x+z=\frac{4}{3}\end{array}\right.$	(d) $\begin{cases}2x-2y=1\\ x+2y=2\\ 2x+3z=6\end{cases}$
(e) $\begin{cases}x+y=6\\ z-3y=7\\ 2x+y+3z=15\end{cases}$	(f) $\begin{cases}2x+y+z=1\\ x+y=3\\ -x+z=5\end{cases}$
(g) $\begin{cases}-2y+z=3\\ -4y+2z=6\\x-2y+z=4\end{cases}$	(h) $\begin{cases}2x+3y-4z=1\\ x-3y+2z=0\\ 3x+2y=5\end{cases}$

Řešení	Ukázat
(a) $[x;y;z]=[20;-13;6]$ v (b) $[x;y;z]=[13;15;17]$ (c) $[x;y;z]=[\frac{11}{12};\frac{7}{12};\frac{5}{12}]$ (d) $[x;y;z]=[1;\frac12;\frac43]$ (e) $[x;y;z]=[\frac{33}4;-\frac94;\frac14]$ (f) $[x;y;z]=[-\frac72;\frac{13}2;\frac32]$ (g) $[x;y;z]=[1;t;2t+3]$ $t\in\mathbb R$ (h) $[x;y;z]=[1;1;1]$

Příklad 5

V $\mathbb{R}^3$ rešte soustavy rovnic:

(a) $\begin{cases}2x+y-z=0\\ 4x+2y+z=0\\ x-y+3z=0\end{cases}$	(b) $\begin{cases}x+y-4z=0\\ x+y+2z=0\\ x+y-z=0\end{cases}$
(c) $\begin{cases}3x+y+z=14\\ -x+2y-3z=-9\\ 5x-y+5z=30\end{cases}$	(d) $\begin{cases}x+2y+3z=6\\ 2x-3y+2z=14\\3x+y-z=-2\end{cases}$
(e) $\begin{cases}5x+5y+z=2\\ 3x-4y-3z=1\\ -2x+y+z=-1\end{cases}$	(f) $\begin{cases}2x+3y+z=15\\ 7x-y+z=9\\ x+2y+z=9\end{cases}$
(g) $\begin{cases}x+y-4z=9\\ x+y+2z=3\\ x+y-z=6\end{cases}$	(h) $\begin{cases}x+y+2z=-1\\ 2x-y+2z=-4\\ 4x+y+4z=-2\end{cases}$
(i) $\begin{cases}3x-2y+4z=1\\ 2x+y-z=3\\ 5x-y+3z=5\end{cases}$	(j) $\begin{cases}2x-3y+4z=1\\ 3x+2y-z=3\\ 4x-y+5z=3\end{cases}$
(k) $\begin{cases}2x-4y-z=1\\ x-5y+z=1\\ 3x-5y-3z=1\end{cases}$	(l) $\begin{cases}\dfrac{y+x}5=\dfrac{z+x}8=\dfrac{x+y}9\\x+y+z=5\end{cases}$
(m) $\begin{cases}x+y-z=0\\x-y+z=2\\-x+y+z=4\end{cases}$	(n) $\begin{cases}5x-2y-z=2\\ 3x+4y-5z=4\\ x+y-2z=-1\end{cases}$

Řešení	Ukázat
(a) $[x;y;z]=[0;0;0]$ (b) $[x;y;z]=[t;-t;0]$ $t\in\mathbb R$ (c) $[x;y;z]=[1;5;6]$ (d) $[x;y;z]=[1;-2;3]$ (e) $[x;y;z]=[1;-1;2]$ (f) $[x;y;z]=[2;4;-1]$ (g) $[x;y;z]=[t,5-t;-1]$ $t\in\mathbb R$ (h) $[x;y;z]=[1;2;-2]$ (i) $[x;y;z]=[t;s;2t+s-3]$ , $t\in\mathbb R$ , $s\in\mathbb R$ (j) $[x;y;z]=[\frac{27}{31};\frac5{31};-\frac2{31}]$ (k) $[x;y;z]=[\frac23;0;\frac13]$ (l) $[x;y;z]=[-5;5;5]$ (m) $[x;y;z]=[1;2;3]$ (n) $9?x;y;z]=[-1;-2;-3]$

Řešení

Ukázat

(a)

$[x;y;z]=[0;0;0]$ (b)

$[x;y;z]=[t;-t;0]$

$t\in\mathbb R$ (c)

$[x;y;z]=[1;5;6]$ (d)

$[x;y;z]=[1;-2;3]$ (e)

$[x;y;z]=[1;-1;2]$ (f)

$[x;y;z]=[2;4;-1]$ (g)

$[x;y;z]=[t,5-t;-1]$

$t\in\mathbb R$ (h)

$[x;y;z]=[1;2;-2]$ (i)

$[x;y;z]=[t;s;2t+s-3]$ ,

$t\in\mathbb R$ ,

$s\in\mathbb R$ (j)

$[x;y;z]=[\frac{27}{31};\frac5{31};-\frac2{31}]$ (k)

$[x;y;z]=[\frac23;0;\frac13]$ (l)

$[x;y;z]=[-5;5;5]$ (m)

$[x;y;z]=[1;2;3]$ (n)

$9?x;y;z]=[-1;-2;-3]$

Příklad 6

V $\mathbb{R}^2$ rešte soustavy rovnic:

(a) $\begin{cases}x+y=1\\ \|y\|-x=1\end{cases}$	(b) $\begin{cases}x+3\|y\|=1\\ x+y+3=0\end{cases}$
(c) $\begin{cases}y-2x=1\\ y-\|x\|=1\end{cases}$	(d) $\begin{cases}\|x\|+y=3\\ x+2y=4\end{cases}$
(e) $\begin{cases}3\|x\|+2y=1\\ 2x-y=3\end{cases}$	(f) $\begin{cases}x+y=2\\ \|3x-y\|=1\end{cases}$
(g) $\begin{cases}\|x-1\|+y=0\\ 2x-y=1\end{cases}$	(h) $\begin{cases}x+2y=6\\ \|x-3\|-y=0\end{cases}$
(i) $\begin{cases}x+2y=2\\ \|2x-3y\|=1\end{cases}$	(j) $\begin{cases}2x-3\|y\|=1\\ \|x\|+2y=4\end{cases}$
(k) $\begin{cases}y-2\|x\|+3=0\\ \|y\|+x-3=0\end{cases}$	(l) $\begin{cases}\|x\|-x+y=10\\ x+\|y\|+y=12\end{cases}$
(m) $\begin{cases}\|x-1\|+\|y-5\|=1\\ y=5+\|x-1\|\end{cases}$	(n) $\begin{cases}\|x-1\|+\|y-2\|=1\\ y=3-\|x-1\|\end{cases}$
(o) $\begin{cases}\|x-3\|+\|y-2\|=3\\ y+\|x-3\|=5\end{cases}$	(p) $\begin{cases}\|2x+3y\|=5\\ \|2x-3y\|=1\end{cases}$
(q) $\begin{cases}2x+y=7\\ \|x-y\|=2\end{cases}$	(r) $\begin{cases}3x-y=1\\ \|x-2y\|=2\end{cases}$

Řešení	Ukázat
(a) $[x;y]=[0;1]$ (b) $[x;y]\in\{[-5;2],[-2;-1]\}$ (c) $[x;y]=[0;1]$ (d) $[x;y]\in\{[-\frac23;\frac73];[2;1]\}$ (e) $[x;y]=[1;-1]$ (f) $[x;y]\in\{[\frac14;\frac74];[\frac34;\frac54]\}$ (g) $[x;y]\in[0;-1]$ (h) $[x;y]\in\{[0;3];[4;1]\}$ (i) $[x;y]\in\{[\frac47;\frac57];[\frac87;\frac37]\}$ (j) $[x;y]=[2;1]$ (k) $[x;y]\in\{[-6;9];[0;-3];[2;1]\}$ (l) $[x;y]=[-\frac85;\frac{34}5]$ (m) $[x;y]\in\{[\frac12;\frac{11}2]; [\frac32;\frac{11}2]\}$ (n) $[x;y]\in\{[t;3-\|t-1\|], 0\le t\le2\}$ (o) $[x;y]\in\{[t;5-\|x-3\|], 0\le t\le6\}$ (p) $[x;y]\in\{[-\frac32;-\frac23];[-1;1];[1;1];[\frac32;\frac23]\}$ (q) $[x;y]\in\{[3;1];[\frac53;\frac{11}3]\}$ (r) $[x;y]\in\{[0;-1];[\frac45;\frac75]\}$

Řešení

Ukázat

(a)

$[x;y]=[0;1]$ (b)

$[x;y]\in\{[-5;2],[-2;-1]\}$ (c)

$[x;y]=[0;1]$ (d)

$[x;y]\in\{[-\frac23;\frac73];[2;1]\}$ (e)

$[x;y]=[1;-1]$ (f)

$[x;y]\in\{[\frac14;\frac74];[\frac34;\frac54]\}$ (g)

$[x;y]\in[0;-1]$ (h)

$[x;y]\in\{[0;3];[4;1]\}$ (i)

$[x;y]\in\{[\frac47;\frac57];[\frac87;\frac37]\}$ (j)

$[x;y]=[2;1]$ (k)

$[x;y]\in\{[-6;9];[0;-3];[2;1]\}$ (l)

$[x;y]=[-\frac85;\frac{34}5]$ (m)

$[x;y]\in\{[\frac12;\frac{11}2]; [\frac32;\frac{11}2]\}$ (n)

$[x;y]\in\{[t;3-|t-1|], 0\le t\le2\}$ (o)

$[x;y]\in\{[t;5-|x-3|], 0\le t\le6\}$ (p)

$[x;y]\in\{[-\frac32;-\frac23];[-1;1];[1;1];[\frac32;\frac23]\}$ (q)

$[x;y]\in\{[3;1];[\frac53;\frac{11}3]\}$ (r)

$[x;y]\in\{[0;-1];[\frac45;\frac75]\}$

(a) $\begin{cases}x+y=1\\ \|y\|-x=1\end{cases}$	(b) $\begin{cases}x+3\|y\|=1\\ x+y+3=0\end{cases}$
(c) $\begin{cases}y-2x=1\\ y-\|x\|=1\end{cases}$	(d) $\begin{cases}\|x\|+y=3\\ x+2y=4\end{cases}$
(e) $\begin{cases}3\|x\|+2y=1\\ 2x-y=3\end{cases}$	(f) $\begin{cases}x+y=2\\ \|3x-y\|=1\end{cases}$
(g) $\begin{cases}\|x-1\|+y=0\\ 2x-y=1\end{cases}$	(h) $\begin{cases}x+2y=6\\ \|x-3\|-y=0\end{cases}$
(i) $\begin{cases}x+2y=2\\ \|2x-3y\|=1\end{cases}$	(j) $\begin{cases}2x-3\|y\|=1\\ \|x\|+2y=4\end{cases}$
(k) $\begin{cases}y-2\|x\|+3=0\\ \|y\|+x-3=0\end{cases}$	(l) $\begin{cases}\|x\|-x+y=10\\ x+\|y\|+y=12\end{cases}$
(m) $\begin{cases}\|x-1\|+\|y-5\|=1\\ y=5+\|x-1\|\end{cases}$	(n) $\begin{cases}\|x-1\|+\|y-2\|=1\\ y=3-\|x-1\|\end{cases}$
(o) $\begin{cases}\|x-3\|+\|y-2\|=3\\ y+\|x-3\|=5\end{cases}$	(p) $\begin{cases}\|2x+3y\|=5\\ \|2x-3y\|=1\end{cases}$
(q) $\begin{cases}2x+y=7\\ \|x-y\|=2\end{cases}$	(r) $\begin{cases}3x-y=1\\ \|x-2y\|=2\end{cases}$

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