# Soustavy lineárních rovnic

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

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

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

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

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

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) $\begin{cases}3x-y=1\\ |x-2y|=2\end{cases}$