Soustava rovnic

1.

$\huge \frac{x+1}{3}-\frac{y+2}{4}=\frac{2(x-y)}{5}$

$\huge \frac{x-3}{4}-\frac{y-3}{3}=2y-x$


$\frac{x+1}{3}-\frac{y+2}{4}=\frac{2(x-y)}{5}\;/\times60$
$\frac{x-3}{4}-\frac{y-3}{3}=2y-x\;/\times12$
$20(x+1)-15(y+2)=24(x-y)$ $3(x-3)-4(y-3)=12(2y-x)$
$20x+20-15y-30=24x-24y$
$3x-9-4y+12=24y-12x$
$-4x+9y=10$
$15x-28y=-3$


$-4x+9y=10\;/\times15$
$-4x+9y=10$
$\underline{15x-28y=-3}\;/\times4$
$-4x+54=10$
$-60x+135y=150$
$-4x=-44$
$\underline{60x-112y=-12}$
$x=11$
$23y=138$
$y=6$
$\large[x;y]=[11;6]$


2.

$\huge\frac{2x-y+3}{3}-\frac{x-2y+3}{4}=4$

$\huge\frac{3x-4y+3}{4}+\frac{4x-2y-9}{3}=4$


$\frac{2x-y+3}{3}-\frac{x-2y+3}{4}=4\;/\times12$
$\frac{3x-4y+3}{4}+\frac{4x-2y-9}{3}=4\;/\times12$
$4(2x-y+3)-3(x-2y+3)=48$ $3(3x-4y+3)+4(4x-2y-9)=48$
$8x-4y+12-3x+6y-9=48$
$9x-12y+9+16x-8y-36=48$
$5x+2y=45$
$25x-20y=75$


$5x+2y=45\;/\times-5$
$5x+2y=45$
$\underline{25x-20y=75}$
$5x+10=45$
$-25x-10y=-225$
$5x=35$
$\underline{25x-20y=75}$
$x=7$
$-30y=-150$

$y=5$
$\large[x;y]=[7;5]$


3.

$\huge\frac{x+1}{y+3}=\frac{1}{2}$

$\huge\frac{x+2}{2y+3}=\frac{1}{3}$


$\frac{x+1}{y+3}=\frac{1}{2}\;/\times2(y+3)$
$\frac{x+2}{2y+3}=\frac{1}{3}\;/\times3(2y+3)$
$2(x+1)=y+3$
$3(x+2)=2y+3$
$2x+2=y+3$
$3x+6=2y+3$
$2x-y=1$
$3x-2y=-3$


$2x-y=1\;/\times-2$
$2x-y=1$
$\underline{3x-2y=-3}$
$10-y=1$
$-4x+2y=-2$
$-y=-9$
$\underline{3x-2y=-3}$
$y=9$
$-x=-5$

$x=5$
$y\neq-3$

$y\neq -\frac{3}{2}$

$\large[x;y]=[5;9]$


4.

$\huge\frac{4}{x-3y}=\frac{7}{9x+2y}$

$\huge\frac{3}{2x+y}=\frac{9}{x-y+1}$


$\frac{4}{x-3y}=\frac{7}{9x+2y}\;/\times(x-3y)(9x+2y)$
$\frac{3}{2x+y}=\frac{9}{x-y+1}$
$4(9x+2y)=7(x-3y)$
$\frac{3}{-2y+y}=\frac{9}{-y-y+1}$
$36x+8y=7x-21y$
$\frac{3}{-y}=\frac{9}{-2y+1}\;/\times(-y)(-2y+1)$
$29x=-29y$
$3(-2y+1)=-9y$
$x=-y$
$-6y+3=-9y$

$y=-1$
$x=1$


$y\neq-\frac{y}{2}$

$y\neq y-1$

$\large[x;y]=[1;-1]$


5.

$\huge x+2y+z=9$

$\huge 2x-3y-z=-12$

$\huge 5x+8y+2z=15$


$x+2y+z=9$
$x=9-2y-z$
$x+z=9$
$2x-3y-z=-12$

$\underline{2x-z=-12}$
$\underline{5x+8y+2z=15}$

$3x=-3$
$2(9-2y-z)-3y-z=-12$

$x=-1$
$\underline{5(9-2y-z)+8y+2z=15}$


$18-4y-2z-3y-z=-12$

$z=9-x-2y$
$\underline{45-10y-5z+8y+2z=15}$

$z=9+1$
$-7y-3z=-30$

$y=10$
$\underline{-2y-3z=-30}\;/\times-1$


$-7y-3z=-30$

$\large[x;y;z]=[-1;0;10]$
$\underline{2y+3z=30}$


$-5y=0$


$y=0$



6.

$\huge x+2y+3z=5$

$\huge 2x-y-z=1$

$\huge x+3y+4z=6$


$x+2y+3z=5$
$x=5-2y-3z$
$2x-y-z=1$

$\underline{x+3y+4z=6}$
$x=6-3y-4z$
$5-2y-3z=6-3y-4z$

$\underline{y=1-z}$

$x+2(1-z)+3z=5$

$\underline{2x-(1-z)-z=1}$

$x+2-2z+3z=5$

$\underline{2x-1+z-z=1}$

$x+z=3$

$x=1\;\;z=2\;\;y=-1$
$\large[x;y;z]=[1;-1;2]$


7.


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