Chapter 1.1, Problem 40E

In the $\left(2\times 2\right)$ 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)

$a_{11}{x}_1+a_{12}{x}_2=b_1$

$a_{21}{x}_1+a_{22}{x}_2=b_2$

(B)

$a_{11}{x}_1+a_{12}{x}_2=b_1$

$\left(a_{21}+c{a}_{11}\right)x_1+\left(a_{22}+c{a}_{12}\right)x_2=b_2+c{b}_1$