In the $(2 \times 2)$ linear systems that follow, the system (B) is obtained from (A) by performing the elementary operation $E_2+c E_1$. Prove that any solution, $x_1=s_1, x_2=s_2$, for (A) is a solution for (B). Similarly, prove that any solution, $x_1=t_1, x_2=t_2$, for (B) is a solution for (A).
(A) $$ \begin{aligned} & a_{11} x_1+a_{12} x_2=b_1 \\ & a_{21} x_1+a_{22} x_2=b_2 \end{aligned} $$
(B) $$ \begin{aligned} & a_{11} x_1+a_{12} x_2=b_1 \\ & \left(a_{21}+c a_{11}\right) x_1+\left(a_{22}+c a_{12}\right) x_2=b_2+c b_1 \end{aligned} $$