$\LARGE ax + by = c$ $\LARGE a'x + b'y = d$ $\LARGE M = \begin{bmatrix} \color{black}a & \color{black}b \\\color{black}a' &\color{black}b'' \end{bmatrix} $ $\LARGE M \begin{bmatrix} \color{black}x \\\ \color{black}y \end{bmatrix} = \begin{bmatrix} \color{black}c \\\ \color{black}d \end{bmatrix}$ $\LARGE det(M) \neq 0 \rightarrow$ $\LARGE \begin{bmatrix} \color{black}x \\\ \color{black}y \end{bmatrix} = M^{-1} \begin{bmatrix} \color{black}c \\\ \color{black}d \end{bmatrix}$

Got it !

Resolution of 2 x 2 equation sytem

$ \LARGE \begin{bmatrix} \color{black}1 & \color{black}2 \\\color{black}2 &\color{black}-1 \end{bmatrix}\cdot \begin{bmatrix} \color{black}x \\\ \color{black}y \end{bmatrix} = \begin{bmatrix} \color{black}4 \\\ \color{black}3 \end{bmatrix} $	$\LARGE \begin{bmatrix} \color{black}x \\\ \color{black}y \end{bmatrix} = \frac{1}{-1-4}\begin{bmatrix} \color{black}-1 & \color{black}-2 \\\ \color{black}-2 & \color{black}1 \end{bmatrix}\cdot \begin{bmatrix} \color{black}4 \\\ \color{black}3 \end{bmatrix}=$ $\LARGE \frac{1}{5}\begin{bmatrix} \color{black}10 \\\ \color{black}5 \end{bmatrix} = \begin{bmatrix} \color{black}2 \\\ \color{black}1 \end{bmatrix}$
$ \LARGE \begin{bmatrix} \color{black}1 & \color{black}-1 \\\color{black}2 &\color{black}3 \end{bmatrix}\cdot \begin{bmatrix} \color{black}x \\\ \color{black}y \end{bmatrix} = \begin{bmatrix} \color{black}1 \\\ \color{black}17 \end{bmatrix} $	$\LARGE \begin{bmatrix} \color{black}x \\\ \color{black}y \end{bmatrix} = \frac{1}{5}\begin{bmatrix} \color{black}3 & \color{black}1 \\\ \color{black}-2 & \color{black}1 \end{bmatrix}\cdot \begin{bmatrix} \color{black}1 \\\ \color{black}17 \end{bmatrix}=$ $\LARGE\begin{bmatrix} \color{black}4 \\\ \color{black}3 \end{bmatrix}$
$ \LARGE \begin{bmatrix} \color{black}\frac{1}{2} & \color{black}5 \\\color{black}2 &\color{black}20 \end{bmatrix}\cdot \begin{bmatrix} \color{black}x \\\ \color{black}y \end{bmatrix} = \begin{bmatrix} \color{black}1 \\\ \color{black}-2 \end{bmatrix} $	impossible , because $\LARGE det(M) = \frac{1}{2}20-2\cdot 5 = 0$