Beispiel Nr: 12
$ \text{Gegeben:} \\ a1 \cdot x +b1 \cdot y =c1\\ a2 \cdot x +b2 \cdot y =c2 \\ \\ \text{Gesucht:} \\\text{x und y} \\ \\ \textbf{Gegeben:} \\ \\ \frac{2}{3} x -\frac{5}{7} y =\frac{2}{3}\\ 1 x +1 y = 10\frac{2}{3} \\ \\ \\ \\ \textbf{Rechnung:} \\\begin{array}{l|l} \begin{array}{l} I \qquad \frac{2}{3} x -\frac{5}{7} y =\frac{2}{3}\\ II \qquad 1 x +1 y = 10\frac{2}{3} \\ \text{I nach y auflösen}\\ \frac{2}{3} x -\frac{5}{7} y =\frac{2}{3} \\ \frac{2}{3} x -\frac{5}{7} y =\frac{2}{3} \qquad /-\frac{2}{3} x\\ -\frac{5}{7} y =\frac{2}{3} -\frac{2}{3} x \qquad /:\left(-\frac{5}{7}\right) \\ y =-\frac{14}{15} +\frac{14}{15} x \\ \text{II nach y auflösen}\\ 1 x +1 y =10\frac{2}{3} \\ 1 x +1 y =10\frac{2}{3} \qquad /-1 x\\ 1 y =10\frac{2}{3} -1 x \qquad /:1 \\ y =10\frac{2}{3} -1 x \\ \text{I = II}\\ -\frac{14}{15} +\frac{14}{15} x =10\frac{2}{3} -1 x \qquad /-\frac{14}{15} x /-10\frac{2}{3} \\ -\frac{14}{15}-10\frac{2}{3} =-1 x -\frac{14}{15} x \\ -11\frac{3}{5} =-1\frac{14}{15} x \qquad /:\left(-1\frac{14}{15}\right) \\ x=6 \\ \text{x in I}\\ y =-\frac{14}{15} +\frac{14}{15} \cdot 6 \\ y=4\frac{2}{3} \\ L=\{6/4\frac{2}{3}\} \end{array} & \begin{array}{l} I \qquad \frac{2}{3} x -\frac{5}{7} y =\frac{2}{3}\\ II \qquad 1 x +1 y = 10\frac{2}{3} \\ \text{I nach x auflösen}\\ \frac{2}{3} x -\frac{5}{7} y =\frac{2}{3} \\ \frac{2}{3} x -\frac{5}{7} y =\frac{2}{3} \qquad /+\frac{5}{7} y\\ \frac{2}{3} x =\frac{2}{3} +\frac{5}{7} y \qquad /:\frac{2}{3} \\ x =1 +1\frac{1}{14} y \\ \text{II nach x auflösen}\\ 1 x +1 y =10\frac{2}{3} \\ 1 x +1 y =10\frac{2}{3} \qquad /-1 y\\ 1 x =10\frac{2}{3} -1 y \qquad /:1 \\ x =10\frac{2}{3} -1 y \\ \text{I = II}\\ 1 +1\frac{1}{14} y =10\frac{2}{3} -1 y \qquad /-1\frac{1}{14} y /-10\frac{2}{3} \\ 1-10\frac{2}{3} =-1 y -1\frac{1}{14} y \\ -9\frac{2}{3} =-2\frac{1}{14} y \qquad /:\left(-2\frac{1}{14}\right) \\ y=4\frac{2}{3} \\ \text{y in I}\\ x =1 +1\frac{1}{14} \cdot 4\frac{2}{3} \\ x=6 \\ L=\{6/4\frac{2}{3}\} \end{array} \end{array} $