Question Number 192992 by gatocomcirrose last updated on 01/Jun/23
$$\begin{cases}{\mathrm{2x}+\mathrm{3y}\equiv\mathrm{1}\left(\mathrm{mod26}\right)}\\{\mathrm{7x}+\mathrm{8y}\equiv\mathrm{2}\left(\mathrm{mod26}\right)}\end{cases} \\ $$$$ \\ $$
Answered by MM42 last updated on 01/Jun/23
$$\mathrm{2}{x}+\mathrm{3}{y}=\mathrm{26}{k}+\mathrm{1}\:\:\&\:\mathrm{7}{x}+\mathrm{8}{y}=\mathrm{26}{k}'+\mathrm{2} \\ $$$$\Rightarrow\mathrm{5}{x}=\mathrm{26}{k}''−\mathrm{2}\Rightarrow{x}\overset{\mathrm{26}} {\equiv}\mathrm{10}\:\checkmark \\ $$$$\Rightarrow\mathrm{2}{x}+\mathrm{3}{y}\overset{\mathrm{26}} {\equiv}\mathrm{20}+\mathrm{3}{y}\overset{\mathrm{26}} {\equiv}\mathrm{1}\Rightarrow\mathrm{3}{y}\overset{\mathrm{26}} {\equiv}−\mathrm{19}\equiv\mathrm{7} \\ $$$$\Rightarrow\mathrm{27}{y}\overset{\mathrm{26}} {\equiv}\mathrm{63}\overset{\mathrm{26}} {\equiv}\mathrm{11}\Rightarrow{y}\overset{\mathrm{26}} {\equiv}\mathrm{11}\:\checkmark \\ $$