Let $R$ be a commutative ring with unity and let $r_0,r_1,b_0,b_1\in R$ such that $\langle b_0,b_1\rangle=(1)$ (that is, the ideal generated by $b_0,b_1$ is the whole ring $R$). Show that the system of equations $t_0b_0+t_1b_1=r_0$, $t_1b_0+t_2b_1=r_1$ admit a solution in $(t_0,t_1,t_2)$ in $R$.
From the hypothesis there exist $(t_0,t_1)$ satisfying the first equation, and there exist $(t_1,t_2)$ satisfying the second equation. But I am having difficulty to show that the $t_1$ obtained from the first equation is a correct choice for $t_1$ in the second equation as well.
