1.
$\huge \frac{x+1}{3}-\frac{y+2}{4}=\frac{2(x-y)}{5}$
$\huge \frac{x-3}{4}-\frac{y-3}{3}=2y-x$
| $\frac{x+1}{3}-\frac{y+2}{4}=\frac{2(x-y)}{5}\;/\times60$
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$\frac{x-3}{4}-\frac{y-3}{3}=2y-x\;/\times12$
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| $20(x+1)-15(y+2)=24(x-y)$ | $3(x-3)-4(y-3)=12(2y-x)$
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| $20x+20-15y-30=24x-24y$
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$3x-9-4y+12=24y-12x$
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| $-4x+9y=10$
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$15x-28y=-3$
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| $-4x+9y=10\;/\times15$
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$-4x+9y=10$
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| $\underline{15x-28y=-3}\;/\times4$
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$-4x+54=10$
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| $-60x+135y=150$
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$-4x=-44$
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| $\underline{60x-112y=-12}$
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$x=11$
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| $23y=138$ |
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| $y=6$
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$\large[x;y]=[11;6]$
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2.
$\huge\frac{2x-y+3}{3}-\frac{x-2y+3}{4}=4$
$\huge\frac{3x-4y+3}{4}+\frac{4x-2y-9}{3}=4$
| $\frac{2x-y+3}{3}-\frac{x-2y+3}{4}=4\;/\times12$
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$\frac{3x-4y+3}{4}+\frac{4x-2y-9}{3}=4\;/\times12$
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| $4(2x-y+3)-3(x-2y+3)=48$ | $3(3x-4y+3)+4(4x-2y-9)=48$
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| $8x-4y+12-3x+6y-9=48$
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$9x-12y+9+16x-8y-36=48$
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| $5x+2y=45$
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$25x-20y=75$
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| $5x+2y=45\;/\times-5$
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$5x+2y=45$
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| $\underline{25x-20y=75}$
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$5x+10=45$
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| $-25x-10y=-225$
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$5x=35$
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| $\underline{25x-20y=75}$
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$x=7$
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| $-30y=-150$
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| $y=5$
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$\large[x;y]=[7;5]$
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3.
$\huge\frac{x+1}{y+3}=\frac{1}{2}$
$\huge\frac{x+2}{2y+3}=\frac{1}{3}$
| $\frac{x+1}{y+3}=\frac{1}{2}\;/\times2(y+3)$
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$\frac{x+2}{2y+3}=\frac{1}{3}\;/\times3(2y+3)$
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| $2(x+1)=y+3$
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$3(x+2)=2y+3$
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| $2x+2=y+3$
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$3x+6=2y+3$
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| $2x-y=1$
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$3x-2y=-3$
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| $2x-y=1\;/\times-2$
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$2x-y=1$
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| $\underline{3x-2y=-3}$
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$10-y=1$
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| $-4x+2y=-2$
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$-y=-9$
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| $\underline{3x-2y=-3}$
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$y=9$
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| $-x=-5$
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| $x=5$
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$y\neq-3$
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$y\neq -\frac{3}{2}$
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$\large[x;y]=[5;9]$
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4.
$\huge\frac{4}{x-3y}=\frac{7}{9x+2y}$
$\huge\frac{3}{2x+y}=\frac{9}{x-y+1}$
| $\frac{4}{x-3y}=\frac{7}{9x+2y}\;/\times(x-3y)(9x+2y)$
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$\frac{3}{2x+y}=\frac{9}{x-y+1}$
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| $4(9x+2y)=7(x-3y)$
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$\frac{3}{-2y+y}=\frac{9}{-y-y+1}$
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| $36x+8y=7x-21y$
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$\frac{3}{-y}=\frac{9}{-2y+1}\;/\times(-y)(-2y+1)$
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| $29x=-29y$
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$3(-2y+1)=-9y$
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| $x=-y$
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$-6y+3=-9y$
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$y=-1$
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| $x=1$
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$y\neq-\frac{y}{2}$
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$y\neq y-1$
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$\large[x;y]=[1;-1]$
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5.
$\huge x+2y+z=9$
$\huge 2x-3y-z=-12$
$\huge 5x+8y+2z=15$
| $x+2y+z=9$
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$x=9-2y-z$
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$x+z=9$
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| $2x-3y-z=-12$
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$\underline{2x-z=-12}$
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| $\underline{5x+8y+2z=15}$
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$3x=-3$
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| $2(9-2y-z)-3y-z=-12$
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$x=-1$
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| $\underline{5(9-2y-z)+8y+2z=15}$
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| $18-4y-2z-3y-z=-12$
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$z=9-x-2y$
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| $\underline{45-10y-5z+8y+2z=15}$
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$z=9+1$
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| $-7y-3z=-30$
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$y=10$
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| $\underline{-2y-3z=-30}\;/\times-1$
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| $-7y-3z=-30$
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$\large[x;y;z]=[-1;0;10]$
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| $\underline{2y+3z=30}$
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| $-5y=0$
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| $y=0$
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6.
$\huge x+2y+3z=5$
$\huge 2x-y-z=1$
$\huge x+3y+4z=6$
| $x+2y+3z=5$
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$x=5-2y-3z$
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| $2x-y-z=1$
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| $\underline{x+3y+4z=6}$
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$x=6-3y-4z$
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| $5-2y-3z=6-3y-4z$
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| $\underline{y=1-z}$
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| $x+2(1-z)+3z=5$
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| $\underline{2x-(1-z)-z=1}$
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| $x+2-2z+3z=5$
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| $\underline{2x-1+z-z=1}$
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| $x+z=3$
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| $x=1\;\;z=2\;\;y=-1$
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$\large[x;y;z]=[1;-1;2]$
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7.