# 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.