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} \\ I \qquad 1\frac{1}{5} x -1\frac{1}{3} y =5\frac{1}{3} \qquad / \cdot2\frac{1}{2}\\ II \qquad 2\frac{1}{2} x -\frac{1}{4} y = 12\frac{3}{8} \qquad / \cdot\left(-1\frac{1}{5}\right)\\ I \qquad 3 x -3\frac{1}{3} y =13\frac{1}{3}\\ II \qquad -3 x +\frac{3}{10} y = -14\frac{17}{20} \\ \text{I + II}\\ I \qquad 3 x -3 x-3\frac{1}{3} y +\frac{3}{10} y =13\frac{1}{3} -14\frac{17}{20}\\ -3\frac{1}{30} y = -1\frac{31}{60} \qquad /:\left(-3\frac{1}{30}\right) \\ y = \frac{-1\frac{31}{60}}{-3\frac{1}{30}} \\ y=\frac{1}{2} \\ \text{y in I}\\ I \qquad 1\frac{1}{5} x -1\frac{1}{3}\cdot \frac{1}{2} =5\frac{1}{3} \\ 1\frac{1}{5} x -\frac{2}{3} =5\frac{1}{3} \qquad / +\frac{2}{3} \\ 1\frac{1}{5} x =5\frac{1}{3} +\frac{2}{3} \\ 1\frac{1}{5} x =6 \qquad / :1\frac{1}{5} \\ x = \frac{6}{1\frac{1}{5}} \\ x=5 \\ 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} \\ I \qquad 1\frac{1}{5} x -1\frac{1}{3} y =5\frac{1}{3} \qquad / \cdot\left(-\frac{1}{4}\right)\\ II \qquad 2\frac{1}{2} x -\frac{1}{4} y = 12\frac{3}{8} \qquad / \cdot1\frac{1}{3}\\ I \qquad -\frac{3}{10} x +\frac{1}{3} y =-1\frac{1}{3}\\ II \qquad 3\frac{1}{3} x -\frac{1}{3} y = 16\frac{1}{2} \\ \text{I + II}\\ I \qquad -\frac{3}{10} x +3\frac{1}{3} x+\frac{1}{3} y -\frac{1}{3} y =-1\frac{1}{3} +16\frac{1}{2}\\ 3\frac{1}{30} x = 15\frac{1}{6} \qquad /:3\frac{1}{30} \\ x = \frac{15\frac{1}{6}}{3\frac{1}{30}} \\ x=5 \\ \text{x in I}\\ I \qquad 1\frac{1}{5} \cdot 5 -1\frac{1}{3}y =5\frac{1}{3} \\ -1\frac{1}{3} y +6 =5\frac{1}{3} \qquad / -6 \\ -1\frac{1}{3} y =5\frac{1}{3} -6 \\ -1\frac{1}{3} y =-\frac{2}{3} \qquad / :\left(-1\frac{1}{3}\right) \\ y = \frac{-\frac{2}{3}}{-1\frac{1}{3}} \\ y=\frac{1}{2} \\ L=\{5/\frac{1}{2}\} \end{array} \end{array}$