1)      Řešte v \Large \mathbb{R} soustavy rovnic a proveďte zkoušky:

\displaystyle a)\quad \underline{\begin{array}{l}4x-y=2\\2x+y=4\end{array}}

\displaystyle b)\quad \underline{\begin{aligned}r+2s &=-1\\3r-2s&=-11\end{aligned}}

\displaystyle c)\quad \underline{\begin{aligned}a+2b&=3\\-a-3b&=-2\end{aligned}}

\displaystyle d)\quad \underline{\begin{aligned}3p-2r&=-1\\p+2r&=-3\end{aligned}}


 

2)      Řešte v \Large \mathbb{R} soustavy rovnic a proveďte zkoušky:

\displaystyle a)\quad \underline{\begin{aligned}2u+5v&=0\\u-v&=7\end{aligned}}

\displaystyle b)\quad \underline{\begin{aligned}-4x-3y&=4\\2x+5y&=12\end{aligned}}

c)\quad \underline{\begin{aligned}6x-5y&=5\\x+y&=-1\end{aligned}}

d)\quad \underline{\begin{aligned}4x-y&=-3\\-12+3y&=9\end{aligned}}


 

3)      Řešte v \Large \mathbb{R} soustavy rovnic a proveďte zkoušky:

\displaystyle a)\quad \underline{\begin{aligned}3x-2y&=2\\2x+5y&=14\end{aligned}}

\displaystyle b)\quad \underline{\begin{aligned}2x+3y&=11\\3x-4y&=25\end{aligned}}

\displaystyle c)\quad \underline{\begin{aligned}4c+5d&=-8\\3c-4d&=25\end{aligned}}

\displaystyle d)\quad \underline{\begin{aligned}2x-3y&=5\\-5x+8y&=-14\end{aligned}}

\displaystyle e)\quad \underline{\begin{aligned}6x-2y&=-6\\9x+7y&=31\end{aligned}}

\displaystyle f)\quad \underline{\begin{aligned}10m+4n&=6\\15m-6n&=15\end{aligned}}

\displaystyle g)\quad \underline{\begin{aligned}u-4v&=7\\9v-2u&=-15\end{aligned}}

\displaystyle h)\quad \underline{\begin{aligned}16a-12b&=-4\\8a+18b&=18\end{aligned}}


 

4)      Řešte v \Large \mathbb{R} soustavy rovnic a proveďte zkoušky:

\displaystyle a)\quad \underline{\begin{aligned}3x+9y&=5\\\frac{x}{3}-\frac{y}{2}&=-\frac{4}{9}\end{aligned}}

\displaystyle b)\quad \underline{\begin{aligned}4x-3y&=8\\\frac{x}{5}+\frac{y}{15}&=-\frac{1}{30}\end{aligned}}

\displaystyle c)\quad \underline{\begin{aligned}\frac{x}{3}+\frac{y}{5}&=0\\\frac{x}{6}-\frac{y}{2}&=\frac{1}{5}\end{aligned}}

\displaystyle d)\quad \underline{\begin{aligned}\frac{x}{3}-\frac{y}{2}&=-\frac{1}{6}\\\frac{x}{4}+\frac{y}{6}&=\frac{1}{5}\end{aligned}}


 

5)      Řešte v \Large \mathbb{R} soustavy rovnic a proveďte zkoušky:

\displaystyle a)\quad \underline{\begin{aligned}2\left( x-1 \right)+3\left( y-4 \right)&=14\\5\left( 6-x \right)-4\left( y-5 \right)&=1\end{aligned}}

\displaystyle b)\quad \underline{\begin{aligned}4\left( u+2 \right)-5\left( v+3 \right)&=-1\\7\left( 2-u \right)-3\left( v+5 \right)&=12\end{aligned}}

\displaystyle c)\quad \underline{\begin{aligned}3\left( r-4 \right)-3\left( s+2 \right)&=2\\4\left( 2-r \right)+2\left( 2s-7 \right)&=14\end{aligned}}

\displaystyle d)\quad \underline{\begin{aligned}2\left( p+2 \right)-3\left( 1-2r \right)&=3\\6\left( 3-p \right)+2\left( 2r+5 \right)&=0\end{aligned}}

\displaystyle e)\quad \underline{\begin{aligned}4\left( 2y-1 \right)+2\left( 3z+4 \right)&=-10\\3\left( 4y+5 \right)-2\left( 3-2z \right)&=3\end{aligned}}

\displaystyle f)\quad \underline{\begin{aligned}3\left( 3s-1 \right)+2\left( 4t+5 \right)&=7\\5\left( 7-6s \right)-3\left( 1-4t \right)&=3\end{aligned}}

\displaystyle g)\quad \underline{\begin{aligned}5\left( x+2 \right)-3\left( 3-2y \right)&=-2\\3\left( 2x-7 \right)+4\left( 4y+5 \right)&=-9\end{aligned}}

\displaystyle h)\quad \underline{\begin{aligned}3\left( 3-a \right)+5\left( b+5 \right)&=13\\2\left( a-3 \right)+3\left( 1-b \right)&=10\end{aligned}}


 

6)      Řešte v \Large \mathbb{R} soustavy rovnic a proveďte zkoušky:

\displaystyle a)\quad \underline{\begin{aligned}0,6x+0,8y&=3,6\\0,9x-0,5y&=0,3\end{aligned}}

\displaystyle b)\quad \underline{\begin{aligned}0,8x+0,5y&=0,4\\0,1x-0,3y&=-1,4\end{aligned}}

\displaystyle c)\quad \underline{\begin{aligned}0,6x+1,5y&=3,6\\\frac{1}{5}x+\frac{1}{2}y&=1\end{aligned}}

\displaystyle d)\quad \underline{\begin{aligned}0,4x+0,5y&=1,5\\\frac{1}{5}x+\frac{1}{4}y&=\frac{3}{4}\end{aligned}}


 

7)      Řešte v \Large \mathbb{R} soustavy rovnic a proveďte zkoušky:

\displaystyle a)\quad \underline{\begin{aligned}x-4y&=1\\\frac{x+2y}{4}-\frac{2x-6y}{3}&=0\end{aligned}}

\displaystyle b)\quad \underline{\begin{aligned}5x+8y&=1\\\frac{x+2y}{3}-\frac{3x-4y}{9}&=\frac{5}{6}\end{aligned}}

\displaystyle c)\quad \underline{\begin{aligned}\frac{x+7y}{4}-\frac{3x+8y}{3}&=1\\\frac{3x+4y}{3}-\frac{4x-5y}{7}&=4\end{aligned}}

\displaystyle d)\quad \underline{\begin{aligned}\frac{x+3y}{7}-\frac{2x+y}{2}&=-3\\\frac{5x-4y}{11}-\frac{6x+5y}{6}&=5\end{aligned}}


 

8)      Řešte v \Large \mathbb{R} soustavy rovnic a proveďte zkoušky:

\displaystyle a)\quad \underline{\begin{aligned}\frac{2r-5}{15}+\frac{3s+20}{10}&=25\\\frac{s-10}{5}-\frac{r-10}{6}&=5\end{aligned}}

\displaystyle b)\quad \underline{\begin{aligned}2u-\frac{v}{3}&=\frac{1}{2}\\\frac{u}{2}+\frac{v}{4}&=1\frac{1}{8}\end{aligned}}

\displaystyle c)\quad \underline{\begin{aligned}\frac{a+b}{2}-\frac{2b}{3}&=2\frac{1}{2}\\1\frac{1}{2}a+2b&=0\end{aligned}}

\displaystyle d)\quad \underline{\begin{aligned}\frac{x-3}{2}-\frac{y-4}{4}&=1\\\frac{2x-5}{3}-\frac{2y-7}{9}&=2\end{aligned}}

\displaystyle e)\quad \underline{\begin{aligned}\frac{c+d}{3}+\frac{d}{5}&=-2\\\frac{2c-d}{3}-\frac{3c}{4}&=1\frac{1}{2}\end{aligned}}

\displaystyle f)\quad \underline{\begin{aligned}\frac{x+7}{2}-3y&=8\\y+3\frac{1}{2}&=\frac{x}{3}\end{aligned}}

\displaystyle g)\quad \underline{\begin{aligned}\frac{v+3}{2}-\frac{z-2}{3}&=2\\\frac{v-1}{4}+\frac{z+1}{3}&=4\end{aligned}}

\displaystyle h)\quad \underline{\begin{aligned}\frac{x}{2}-\frac{y}{3}&=1\\\frac{5}{6}\left( 3x-1 \right)&=\frac{3}{4}\left( 2y-5 \right)\end{aligned}}


 

9)      Řešte v \Large \mathbb{R} soustavy rovnic a proveďte zkoušky:

\displaystyle a)\quad \underline{\begin{aligned}\frac{6-2x}{2}-\frac{y+5}{5}&=3\\{{\left( x-2 \right)}^{2}}-{{\left( x+3 \right)}^{2}}&=2y+5\end{aligned}}

\displaystyle b)\quad \underline{\begin{aligned}\left( p-r \right)\left( p+r \right)&={{\left( p-1 \right)}^{2}}-{{\left( r-1 \right)}^{2}}\\{{p}^{2}}+{{r}^{2}}&={{\left( p-1 \right)}^{2}}+{{\left( r-1 \right)}^{2}}\end{aligned}}

\displaystyle c)\quad \underline{\begin{aligned}\frac{2p}{7}+\frac{q}{2}&=\frac{1}{14}\\{{\left( p+3 \right)}^{2}}-{{\left( p+1 \right)}^{2}}&=4\left( 3-q \right)\end{aligned}}

\displaystyle d)\quad \underline{\begin{aligned}{{m}^{2}}+{{n}^{2}}&={{\left( m+1 \right)}^{2}}+{{\left( n+1 \right)}^{2}}\\{{m}^{2}}+{{n}^{2}}&={{\left( m-1 \right)}^{2}}+{{\left( n-1 \right)}^{2}}\end{aligned}}