Beispiel Nr: 09
$ \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{1}{2} x +1 y =2\\ \frac{1}{2} x -3 y = -3 \\ \\ \\ \\ \textbf{Rechnung:} \\\begin{array}{l|l} \begin{array}{l} \\I \qquad -\frac{1}{2} x +1 y =2\\ II \qquad \frac{1}{2} x -3 y = -3 \\ I \qquad -\frac{1}{2} x +1 y =2 \qquad / \cdot\left(-\frac{1}{2}\right)\\ II \qquad \frac{1}{2} x -3 y = -3 \qquad / \cdot\left(-\frac{1}{2}\right)\\ I \qquad \frac{1}{4} x -\frac{1}{2} y =-1\\ II \qquad -\frac{1}{4} x +1\frac{1}{2} y = 1\frac{1}{2} \\ \text{I + II}\\ I \qquad \frac{1}{4} x -\frac{1}{4} x-\frac{1}{2} y +1\frac{1}{2} y =-1 +1\frac{1}{2}\\ 1 y = \frac{1}{2} \qquad /:1 \\ y = \frac{\frac{1}{2}}{1} \\ y=\frac{1}{2} \\ \text{y in I}\\ I \qquad -\frac{1}{2} x +1\cdot \frac{1}{2} =2 \\ -\frac{1}{2} x +\frac{1}{2} =2 \qquad / -\frac{1}{2} \\ -\frac{1}{2} x =2 -\frac{1}{2} \\ -\frac{1}{2} x =1\frac{1}{2} \qquad / :\left(-\frac{1}{2}\right) \\ x = \frac{1\frac{1}{2}}{-\frac{1}{2}} \\ x=-3 \\ L=\{-3/\frac{1}{2}\} \end{array} & \begin{array}{l} \\I \qquad -\frac{1}{2} x +1 y =2\\ II \qquad \frac{1}{2} x -3 y = -3 \\ I \qquad -\frac{1}{2} x +1 y =2 \qquad / \cdot\left(-3\right)\\ II \qquad \frac{1}{2} x -3 y = -3 \qquad / \cdot\left(-1\right)\\ I \qquad 1\frac{1}{2} x -3 y =-6\\ II \qquad -\frac{1}{2} x +3 y = 3 \\ \text{I + II}\\ I \qquad 1\frac{1}{2} x -\frac{1}{2} x-3 y +3 y =-6 +3\\ 1 x = -3 \qquad /:1 \\ x = \frac{-3}{1} \\ x=-3 \\ \text{x in I}\\ I \qquad -\frac{1}{2} \cdot \left(-3\right) +1y =2 \\ 1 y +1\frac{1}{2} =2 \qquad / -1\frac{1}{2} \\ 1 y =2 -1\frac{1}{2} \\ 1 y =\frac{1}{2} \qquad / :1 \\ y = \frac{\frac{1}{2}}{1} \\ y=\frac{1}{2} \\ L=\{-3/\frac{1}{2}\} \end{array} \end{array} $