$\text{a)}$  

$\{\begin{array}{l}10\left(y+2\right)+x-5=x-5\\ 25\left(y+2\right)+3\left(x-5\right)=-9\end{array}$  

$\{\begin{array}{l}10\left(y+2\right)=0\mid :10\\ 25\left(y+2\right)+3\left(x-5\right)=-9\end{array}$  

$\{\begin{array}{l}y+2=0\\ 25\left(y+2\right)+3\left(x-5\right)=-9\end{array}$  

$\{\begin{array}{l}y=-2\\ 25\left(y+2\right)+3\left(x-5\right)=-9\end{array}$  

Podstawiamy y=-2 do drugiego równania.

$\{\begin{array}{l}y=-2\\ 25\left(-2+2\right)+3\left(x-5\right)=-9\end{array}$  

$\{\begin{array}{l}y=-2\\ 25\cdot 0+3\left(x-5\right)=-9\end{array}$  

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

$\{\begin{array}{l}y=-2\\ x-5=-3\end{array}$  

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

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

---

$\text{b)}$  

$\{\begin{array}{l}\frac{x+2y}{4}-\left(x-\frac{1}{3}y\right)=1+y\\ \frac{y+3}{6}+\frac{2-x}{8}=\frac{3}{2}\mid \cdot 24\end{array}$  

$\{\begin{array}{l}\frac{x}{4}+\frac{y}{2}-x+\frac{1}{3}y=1+y\\ 4\left(y+3\right)+3\left(2-x\right)=36\end{array}$  

$\{\begin{array}{l}-\frac{3}{4}x+\frac{3}{6}y+\frac{2}{6}y-\frac{6}{6}y=1\\ 4y+12+6-3x=36\end{array}$  

$\{\begin{array}{l}-\frac{3}{4}x-\frac{1}{6}y=1\mid \cdot 12\\ -3x+4y=18\end{array}$  

$\{\begin{array}{l}-9x-2y=12\mid \cdot 2\\ -3x+4y=18\end{array}$  

$\{\begin{array}{l}-18x-4y=24\\ -3x+4y=18\end{array}$  

Dodajemy równania stronami.

$-21x=42\mid :\left(-21\right)$  

$x=-2$  

Wartość x=-2 podstawiamy do dowolnego równania układu i obliczamy y.

$\{\begin{array}{l}x=-2\\ -3x+4y=18\end{array}$  

$\{\begin{array}{l}x=-2\\ -3\cdot \left(-2\right)+4y=18\end{array}$  

$\{\begin{array}{l}x=-2\\ 6+4y=18\end{array}$  

$\{\begin{array}{l}x=-2\\ 4y=12\mid :4\end{array}$  

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

---

$\text{c)}$  

$\{\begin{array}{l}{\left(x-1\right)}^2-{\left(y-3\right)}^2=\left(x-y\right)\left(x+y\right)\\ \frac{x+4}{2}-\frac{y+4}{3}=1\mid \cdot 6\end{array}$  

$\{\begin{array}{l}{x}^2-2x+1-\left(y^2-6y+9\right)=x^2-y^2\\ 3\left(x+4\right)-2\left(y+4\right)=6\end{array}$  

$\{\begin{array}{l}-2x+1-y^2+6y-9=-y^2\\ 3x+12-2y-8=6\end{array}$  

$\{\begin{array}{l}-2x+6y=8\\ 3x-2y=2\mid \cdot 3\end{array}$  

$\{\begin{array}{l}-2x+6y=8\\ 9x-6y=6\end{array}$  

Dodajemy równania stronami.

$7x=14\mid :7$  

$x=2$  

Wartość x=2 podstawiamy do dowolnego równania układu i obliczamy y.

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

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

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

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

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

---

$\text{d)}$  

$\{\begin{array}{l}{\left(x-2\right)}^2-{\left(y-1\right)}^2=\left(x-y\right)\left(x+y\right)+5\\ 2\left(x-y\right)+y^2=\left(y-2\sqrt{2}\right)\left(y+2\sqrt{2}\right)\end{array}$  

$\{\begin{array}{l}{x}^2-4x+4-\left(y^2-2y+1\right)=x^2-y^2+5\\ 2x-2y+y^2=y^2-8\end{array}$  

$\{\begin{array}{l}-4x+4-y^2+2y-1=-y^2+5\\ 2x-2y=-8\end{array}$  

$\{\begin{array}{l}-4x+2y=2\\ 2x-2y=-8\end{array}$  

Dodajemy równania stronami.

$-2x=-6\mid :\left(-2\right)$  

$x=3$  

Wartość x=3 podstawiamy do dowolnego równania układu i obliczamy y.

$\{\begin{array}{l}x=3\\ -4x+2y=2\end{array}$  

$\{\begin{array}{l}x=3\\ -4\cdot 3+2y=2\end{array}$  

$\{\begin{array}{l}x=3\\ -12+2y=2\end{array}$  

$\{\begin{array}{l}x=3\\ 2y=14\mid :2\end{array}$  

$\{\begin{array}{l}x=3\\ y=7\end{array}$