Beispiel Nr: 21
$ \text{Gegeben:} \\ a1 \cdot x +b1 \cdot y =c1\\ a2 \cdot x +b2 \cdot y =c2 \\ \\ \text{Gesucht:} \\\text{x und y} \\ \\ \textbf{Gegeben:} \\ \\ 2x -7y =-8\\ 7x -1y = -9 \\ \\ \\ \\ \textbf{Rechnung:} \\\begin{array}{l|l} \begin{array}{l} I \qquad 2 x -7 y =-8\\ II \qquad 7 x -1 y = -9 \\ \text{I nach x auflösen}\\ 2 x -7 y =-8 \\ 2 x -7 y =-8 \qquad /+7 y\\ 2 x =-8 +7 y \qquad /:2 \\ x =-4 +3\frac{1}{2} y \\ \text{I in II}\\ 7 (-4 +3\frac{1}{2} y ) + -1 y = -9 \\ -28 +24\frac{1}{2} y -1 y = -9 \qquad / -\left(-28\right) \\ +24\frac{1}{2} y -1 y = -9 -\left(-28\right) \\ 23\frac{1}{2} y = 19 \qquad /:23\frac{1}{2} \\ y = \frac{19}{23\frac{1}{2}} \\ y=\frac{38}{47} \\ x =-4 +3\frac{1}{2} y \\ x =-4 +3\frac{1}{2} \cdot \frac{38}{47} \\ x=-1\frac{8}{47} \\ L=\{-1\frac{8}{47}/\frac{38}{47}\} \end{array} & \begin{array}{l} I \qquad 2 x -7 y =-8\\ II \qquad 7 x -1 y = -9 \\ \text{I nach y auflösen}\\ 2 x -7 y =-8 \\ 2 x -7 y =-8 \qquad /-2 x\\ -7 y =-8 -2x \qquad /:\left(-7\right) \\ y =1\frac{1}{7} +\frac{2}{7}x \\ \text{I in II}\\ 7x + -1(1\frac{1}{7} +\frac{2}{7} x ) = -9 \\ -1\frac{1}{7} -\frac{2}{7} x -1 x = -9 \qquad / -\left(-1\frac{1}{7}\right) \\ -\frac{2}{7} x -1 x = -9 -\left(-1\frac{1}{7}\right) \\ 6\frac{5}{7} x = -7\frac{6}{7} \qquad /:6\frac{5}{7} \\ x = \frac{-7\frac{6}{7}}{6\frac{5}{7}} \\ x=-1\frac{8}{47} \\ y =1\frac{1}{7} +\frac{2}{7} x \\ y =1\frac{1}{7} +\frac{2}{7} \cdot \left(-1\frac{8}{47}\right) \\ y=\frac{38}{47} \\ L=\{-1\frac{8}{47}/\frac{38}{47}\} \end{array} \end{array} $