4.6 KiB
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Soustavy rovnic
Příklady
substituční
první $x+2y=5$
druhá 2x+y=7
první $x=5-2y$
druhá 2(5-2y)+y=7 | 10-4y+y=7 | 10-3y=7 | -3y=-3 | 3y=3 | y=1
druhá $y=7-2x$
první x+2(7-2x)=5|x+14-4x=5|14-3x=5|-3x=-9|x=3
$y=1$
x=3
Sčítací
první $2x-y=12$
druhá 3x+2y=25
první $6x-3y=36$
druhá $-6x+4y=-50$
výsledek y=-14
první $4x-2y=24$
druhá $3x+2y=25$
výsledek 7x=49|x=7
DÚ 28.3.2022
A (sčítací)
$3x-\frac4{y-2}=13$
5x+3(y+2)=20
$3x-\frac4{y-2}=13$
$3x(y-2)-4=13(y-2)$
$3xy-6x-4=13y-26$
3xy-19y=30
$5x+3(y+2)=20$
5x+3y=14
$9y^2+5y-26=0$
$D=b^2-4ac$
$D=5^2-49(-26)$
$D=25+936$
$D=961$
$\sqrt{D}=31$
$y_{1;2}=\frac{-b\pm\sqrt{D}}{2a}$
$y_1=\frac{-5+31}{2*9}=\frac{13}9$
y_2=\frac{-5-31}{2*9}=-2
$5x+3(y+2)=20$
$5x+3(-2+2)=20$
$5x=20$
x=4
$5x+3(y+2)=20$
$5x+3(\frac{13}9+2)=20$
$5x+\frac{39}9+2=20$
$5x+6+\frac13=20$
$5x+\frac13=14$
$5x=14-\frac13$
x=\frac{41}{15}
B (substituční)
$\frac{x-1}8-\frac{y-7}4=1$
2x+\frac{2y-5}3 = 21
$x-1-2{y-7}=8$
$x-8-2y=8$
$x-2y=16$
$x=16+2y$
$x=16+2(34-3x)$
$x=16+2(34-3(16+2(34-3(16+2y))))$
$x=16+2(\frac{14}5)$
$x=16+\frac{28}5$
$x=16+5.6$
x=21.6
$2x+\frac{2y-5}3 = 21$
$6x+2y-5 = 63$
$6x+2y=68$
$2y=68-6x$
$y=34-3(16+2y)$
$y=34-48+6y$
$y=-14+6y$
$-5y=-14$
$5y=14$
y=\frac{14}5
C (srovnávací)
$\frac1{x-2}+\frac1{y+3}=18$
\frac1{x-2}-\frac1{y+3}=6
$18-\frac1{y+3}=6+\frac1{y+3}$
$18(y+3)-1=6(y+3)+1$
$18y+54-1=6y+18+1$
$18y+53=6y+19$
$18y=6y-34$
$9y=3y-17$
$6y=-17$
y=-\frac{17}6
D
$\frac3{x+1}+\frac2{y-2}=-1$
\frac1{x+1}+\frac1{y-2}=-\frac23
$x^2+y^2=10$
x+2y=1
x=1-2y
$(1-2y)^2+y^2=10$
$1-4y+4y^2+y^2=10$
$1-4y+5y^2=10$
5y^2-4y-9=0
$D=16+180=196$
\sqrt{D}=14
$y_{1;2}=\frac{-b\pm\sqrt{D}}{2a}$
$y_1 = \frac{4+14}{10}=\frac{18}{10}=1.8$
y_2=\frac{4-14}{10}=-1
$x^2+y^2=8$
x+3y=4
x=4-3y
$(4-3y)^2+y^2=8$
$16-24y+9y^2+y^2=8$
$16-24y+10y^2=8$
10y^2-24y+8=0
$D=b^2 - 4ac$
$D=24^2-320$
$D=576-320$
$D=256$
\sqrt{D}=16
$x_{1;2}=\frac{-b\pm\sqrt{D}}{2a}$
$x_1=\frac{24+16}{2*10}=\frac{40}{20}=2$
x_2=\frac{24-16}{2*10}=\frac{8}{20}=0.4
$2x-3y+4z=8$
$3x-2y-z=-4$
-4x+5y-2z=0
$3x+5y+z=9$
$-2x+2y+z=0$
10x-10y-3z=z-5y
10x-5y-4z=0
$3x+5y+z=9$
$+2x-2y-z=0$
5x+3y=9
$12x+20y+4z=36$
$10x-5y-4z=0$
22x+15y=36
$25x+15y=45$
$-22x-15y=-36$
$3x=9$
x=3
$22*3+15y=36$
$66+15y=36$
$15y=-30$
y=-2
$33+5(-2)+z=9$
$9-10+z=9$
$-10+z=0$
z=10
K=\{[3,-2,10]\}
$3x+5y+z=9$
$-2x+2y+z=0$
5x-19y+5z=27
$3x+5y+z=9$
$2x-2y-z=0$
5x+3y=9
$-15x-25y-5z=-45$
$5x-19y+5z=27$
-10x-44y=-18
$10x+6y=18$
$-10x-44y=-18$
$-38y=0$
y=0
$10x=18$
x=1.8
$3x+5y+z=9$
$31.8+50+z=9$
$5.4+z=9$
z=3.6
K=\{[1.8, 0, 3.6]\}
$3x+5y+z=9$
$-1x+1y+z=0$
5x+11y+z=27
$4x-3y+2z=9$
$3x+2y-2z=2$
-7x-16y+14z=6
$4x-3y+2z=9$
$3x+2y-2z=2$
7x-y=11
$-28x+21y-14z=-63$
$-7x-16y+14z=6$
-35x+5y=-57
$-35x+5y=-55$
-35x+5y=-57
$4x-3y+2z=9$
$6x+4y-4z=4$
-7x-16y+14z=8
$8x-6y+4z=18$
$6x+4y-4z=4$
14x-2y=22
$-28x+21y-14z=-63$
$-7z-16y+14z=8$
-35x+5y=-55
$70x-25y=110$
-70x+25y=-110
Na dvorku pobíhají zvířátka: králící a kuřata Pepík je pozoroval a napočítal celkem 30 hlaviček a 88 nožiček. Kolik je králíčků a kolik kuřátek?
$x4+(30-x)2=88$
$x4+302-x*2=88$
$4x-2x=28$
$2x=28$
x=14 - králíčci
(30-x)=16 - kuřátka
$x+y=12$
$x+z=19$
y+z=17
$x+y=12$
$x+z=19$
y-z=-7
$y+z=17$
-y+z=7
$2z=24$
z=12
$y+z=17$
$y+12=17$
y=5
$x+y=12$
$x+5=12$
x=7
$2x+3y=29$
$3x+4z=44$
2y+3z=38
$6x+9y=87$
$6x+8z=88$
9y-8z=-1
$18y-16z=-2$
18y+27z=342
z=344/43=8
$4x+2y+z=24$
$y-2x=-6$
z+3x=11
$3y-6x=-18$
2z+6x=22
$x+2y-z=6$
$2x+4y-2z=12$
-7x-14y+7z=-42
$-z=6-2y-x$
z=x+2y-6
K=\{[x;y;z]: [x;y;x+2y-6]\}; x;y\in\mathbb{R}\}
$2x-y+z=3$
$3x+y-2z=-1$
x-3y+4z=7
$2x-y+z=3$
$3x+y-2z=-1$
5x-z=2
$6x+2y-4z=-2$
$x-3y+4z=7$
7x-y=5
$35x-7z=14$
$35x-5y=25$
$2y=11$
y=10.5
$2x+y-3z=6$
$x-y+7z=11$
8x+y+5z=50
$2x+y-3z=6$
$x-y+7z=11$
3x+4z=17
$5x-5y+35z=55$
$56x+7y+35z=350$
51x+12y=295
$153x+204z=867$
$153x+36y=885$
34z-6y=-3
$x+y+z+t=6$
$x-y-z+2t=2$
$2x+2y+z+t=8$
x+y-z-t=-2
$x+y+z+t=6$
$x-y-z+2t=2$
2x+3t=8
$x+y+z+t=6$
$x+y-z-t=-2$
$2x+2y=4$
x+y=2
$2x+3t=8$
$2x+2y=4$
3t-2y=4
$x+y+z+t=6$
$2x+2y+z+t=8$
x+y=2
$x+y+z+t=6$
$x+y-z-t=-2$
$2z+2t=8$
z+t=4
$x+y+z+t=6$
$x-y-z+2t=2$
2y+2z-t=4
$x-y-z+2t=2$
$x+y-z-t=-2$
x-2z+t=0
$2x+2y+z+t=8$
$x+y-z-t=-2$
$3x+3y=6$
x+y=2
$x+y-z-t=-2$
$x+y+z+t=6$
$2x+2y=4$
x+y=2
$2y+2z-t=4$
$x-2z+t=0$
2y+x=4
$x+y=2$
$2y+x=4$
$y=2$
x=0