Beispiel Nr: 11
$ \text{Gegeben:} \\ a1 \cdot x +b1 \cdot y =c1\\ a2 \cdot x +b2 \cdot y =c2 \\ \\ \text{Gesucht:} \\\text{x und y} \\ \\ \textbf{Gegeben:} \\ \\ 1\frac{1}{5} x -1\frac{1}{3} y =5\frac{1}{3}\\ 2\frac{1}{2} x -\frac{1}{4} y = 12\frac{3}{8} \\ \\ \\ \\ \textbf{Rechnung:} \\\begin{array}{l|l} \begin{array}{l} I \qquad 1\frac{1}{5} x -1\frac{1}{3} y =5\frac{1}{3}\\ II \qquad 2\frac{1}{2} x -\frac{1}{4} y = 12\frac{3}{8} \\ \text{I nach y auflösen}\\ 1\frac{1}{5} x -1\frac{1}{3} y =5\frac{1}{3} \\ 1\frac{1}{5} x -1\frac{1}{3} y =5\frac{1}{3} \qquad /-1\frac{1}{5} x\\ -1\frac{1}{3} y =5\frac{1}{3} -1\frac{1}{5} x \qquad /:\left(-1\frac{1}{3}\right) \\ y =-4 +\frac{9}{10} x \\ \text{II nach y auflösen}\\ 2\frac{1}{2} x -\frac{1}{4} y =12\frac{3}{8} \\ 2\frac{1}{2} x -\frac{1}{4} y =12\frac{3}{8} \qquad /-2\frac{1}{2} x\\ -\frac{1}{4} y =12\frac{3}{8} -2\frac{1}{2} x \qquad /:\left(-\frac{1}{4}\right) \\ y =-49\frac{1}{2} +10 x \\ \text{I = II}\\ -4 +\frac{9}{10} x =-49\frac{1}{2} +10 x \qquad /-\frac{9}{10} x /+49\frac{1}{2} \\ -4+49\frac{1}{2} =10 x -\frac{9}{10} x \\ 45\frac{1}{2} =9\frac{1}{10} x \qquad /:9\frac{1}{10} \\ x=5 \\ \text{x in I}\\ y =-4 +\frac{9}{10} \cdot 5 \\ y=\frac{1}{2} \\ L=\{5/\frac{1}{2}\} \end{array} & \begin{array}{l} I \qquad 1\frac{1}{5} x -1\frac{1}{3} y =5\frac{1}{3}\\ II \qquad 2\frac{1}{2} x -\frac{1}{4} y = 12\frac{3}{8} \\ \text{I nach x auflösen}\\ 1\frac{1}{5} x -1\frac{1}{3} y =5\frac{1}{3} \\ 1\frac{1}{5} x -1\frac{1}{3} y =5\frac{1}{3} \qquad /+1\frac{1}{3} y\\ 1\frac{1}{5} x =5\frac{1}{3} +1\frac{1}{3} y \qquad /:1\frac{1}{5} \\ x =4\frac{4}{9} +1\frac{1}{9} y \\ \text{II nach x auflösen}\\ 2\frac{1}{2} x -\frac{1}{4} y =12\frac{3}{8} \\ 2\frac{1}{2} x -\frac{1}{4} y =12\frac{3}{8} \qquad /+\frac{1}{4} y\\ 2\frac{1}{2} x =12\frac{3}{8} +\frac{1}{4} y \qquad /:2\frac{1}{2} \\ x =4\frac{19}{20} +\frac{1}{10} y \\ \text{I = II}\\ 4\frac{4}{9} +1\frac{1}{9} y =4\frac{19}{20} +\frac{1}{10} y \qquad /-1\frac{1}{9} y /-4\frac{19}{20} \\ 4\frac{4}{9}-4\frac{19}{20} =\frac{1}{10} y -1\frac{1}{9} y \\ -0,506 =-1\frac{1}{90} y \qquad /:\left(-1\frac{1}{90}\right) \\ y=\frac{1}{2} \\ \text{y in I}\\ x =4\frac{4}{9} +1\frac{1}{9} \cdot \frac{1}{2} \\ x=5 \\ L=\{5/\frac{1}{2}\} \end{array} \end{array} $