a)

$\{\begin{array}{l}\frac{x}{2}-\frac{y}{3}=1\mid \cdot 6\\ x+2y=10\mid \cdot \left(-3\right)\end{array}$  

 

$\{\begin{array}{l}3x-2y=6\\ -3x-6y=-30\end{array}$  

$+\underline{}$  

$3x-2y-3x-6y=6-30$  

$-8y=-24\mid :\left(-8\right)$  

$y=3$  

 

$x+2y=10$  

$x+2\cdot 3=10$  

$x+6=10$   

$x=4$          

 

$\{\begin{array}{l}x=4\\ y=3\end{array}$  

 

---

b)

$\{\begin{array}{l}\frac{x}{2}+\frac{y}{4}=1\mid \cdot 4\\ \frac{5x}{9}+\frac{y}{3}=1\mid \cdot 9\end{array}$  

 

$\{\begin{array}{l}2x+y=4\mid \cdot \left(-3\right)\\ 5x+3y=9\end{array}$  

 

$\{\begin{array}{l}-6x-3y=-12\\ 5x+3y=9\end{array}$  

$+\underline{}$  

$-6x-3y+5x+3y=-12+9$  

$-x=-3$  

$x=3$  

 

$2x+y=4$  

$2\cdot 3+y=4$  

$6+y=4$  

$y=-2$  

 

$\{\begin{array}{l}x=3\\ y=-2\end{array}$  

 

c)

$\{\begin{array}{l}\frac{x-2}{5}-\frac{y-2}{5}=0\mid \cdot 5\\ \frac{x}{12}+\frac{y}{12}=\frac{1}{3}\mid \cdot 12\end{array}$  

 

$\{\begin{array}{l}x-2-\left(y-2\right)=0\\ x+y=4\end{array}$  

 

$\{\begin{array}{l}x-2-y+2=0\\ x+y=4\end{array}$  

 

$\{\begin{array}{l}x-y=0\\ x+y=4\end{array}$  

 

$\{\begin{array}{l}x=y\\ y+y=4\end{array}$  

 

$\{\begin{array}{l}x=y\\ 2y=4\mid :2\end{array}$  

 

$\{\begin{array}{l}x=y\\ y=2\end{array}$  

 

$\{\begin{array}{l}x=2\\ y=2\end{array}$  

 

 

---

d)

$\{\begin{array}{l}\frac{x-1}{2}=1-\frac{y-1}{3}\mid \cdot 6\\ \frac{y-1}{2}=1-\frac{x-1}{3}\mid \cdot 6\end{array}$  

 

$\{\begin{array}{l}3\left(x-1\right)=6-2\left(y-1\right)\\ 3\left(y-1\right)=6-2\left(x-1\right)\end{array}$  

 

$\{\begin{array}{l}3x-3=6-2y+2\\ 3y-3=6-2x+2\end{array}$  

 

$\{\begin{array}{l}3x+2y=11\mid \cdot 2\\ 2x+3y=11\mid \cdot \left(-3\right)\end{array}$  

 

$\{\begin{array}{l}6x+4y=22\\ -6x-9y=-33\end{array}$  

$+\underline{}$  

$6x+4y-6x-9y=22-33$  

$-5y=-11$  

$y=\frac{11}{5}$  

 

$6x+4y=22$  

$6x+4\cdot \frac{11}{5}=22$  

$6x+\frac{44}{5}=22$  

$6x+8,8=22$  

$6x=13,2\mid :6$  

$x=2,2$  

 

$\{\begin{array}{l}x=2,2\\ y=\frac{11}{5}=2,2\end{array}$