Consider the system of two linear equations with two variables \(x\) and \(y\):
\[ \begin{cases} a_1x + b_1y = c_1 \\ a_2x + b_2y = c_2 \end{cases}, \quad \frac{a_1}{a_2} \neq \frac{b_1}{b_2} \]
Solution for \(x\):
\[ x = \frac{ \begin{vmatrix} c_1 & b_1 \\ c_2 & b_2 \end{vmatrix} }{ \begin{vmatrix} a_1 & b_1 \\ a_2 & b_2 \end{vmatrix} } = \frac{c_1 b_2 - c_2 b_1}{a_1 b_2 - a_2 b_1} \]
Solution for \(y\):
\[ y = \frac{ \begin{vmatrix} a_1 & c_1 \\ a_2 & c_2 \end{vmatrix} }{ \begin{vmatrix} a_1 & b_1 \\ a_2 & b_2 \end{vmatrix} } = \frac{a_1 c_2 - a_2 c_1}{a_1 b_2 - a_2 b_1} \]
