$\begin{array}{l} \text{Gegeben:} \\ a1 \cdot x +b1 \cdot y =c1\\ a2 \cdot x +b2 \cdot y =c2 \\ \\ \text{Gesucht:} \\\text{x und y} \\ \\ \textbf{Gegeben:} \\ \\ 4x -5y =43\\ -3x -7y = 0 \\ \\ \\ \\ \textbf{Rechnung:} \\\begin{array}{l|l} \begin{array}{l} I \qquad 4 x -5 y =43\\ II \qquad -3 x -7 y = 0 \\ \text{I nach x auflösen}\\ 4 x -5 y =43 \\ 4 x -5 y =43 \qquad /+5 y\\ 4 x =43 +5 y \qquad /:4 \\ x =10\frac{3}{4} +1\frac{1}{4} y \\ \text{I in II}\\ -3 (10\frac{3}{4} +1\frac{1}{4} y ) + -7 y = 0 \\ -32\frac{1}{4} -3\frac{3}{4} y -7 y = 0 \qquad / -\left(-32\frac{1}{4}\right) \\ -3\frac{3}{4} y -7 y = 0 -\left(-32\frac{1}{4}\right) \\ -10\frac{3}{4} y = 32\frac{1}{4} \qquad /:\left(-10\frac{3}{4}\right) \\ y = \frac{32\frac{1}{4}}{-10\frac{3}{4}} \\ y=-3 \\ x =10\frac{3}{4} +1\frac{1}{4} y \\ x =10\frac{3}{4} +1\frac{1}{4} \cdot \left(-3\right) \\ x=7 \\ L=\{7/-3\} \end{array} & \begin{array}{l} I \qquad 4 x -5 y =43\\ II \qquad -3 x -7 y = 0 \\ \text{I nach y auflösen}\\ 4 x -5 y =43 \\ 4 x -5 y =43 \qquad /-4 x\\ -5 y =43 -4x \qquad /:\left(-5\right) \\ y =-8\frac{3}{5} +\frac{4}{5}x \\ \text{I in II}\\ -3x + -7(-8\frac{3}{5} +\frac{4}{5} x ) = 0 \\ 60\frac{1}{5} -5\frac{3}{5} x -7 x = 0 \qquad / -60\frac{1}{5} \\ -5\frac{3}{5} x -7 x = 0 -60\frac{1}{5} \\ -8\frac{3}{5} x = -60\frac{1}{5} \qquad /:\left(-8\frac{3}{5}\right) \\ x = \frac{-60\frac{1}{5}}{-8\frac{3}{5}} \\ x=7 \\ y =-8\frac{3}{5} +\frac{4}{5} x \\ y =-8\frac{3}{5} +\frac{4}{5} \cdot 7 \\ y=-3 \\ L=\{7/-3\} \end{array} \end{array} \end{array}$