8. Find all non-negative solutions and all positive solutions of the following equations:
(i) $5 x_{1}+7 x_{2}=41$;
(ii) $96 x_{1}+97 x_{2}=1000$;
(iii) $7 x_{1}+3 x_{2}=123$;
(iv) $15 x_{1}+12 x_{2}+20 x_{3}=59$.
8. (i) $x_{1}=4+7 t, x_{2}=3-5 t$. $x_{1}=4, x_{2}=3$.
(ii) $x_{1}=-1000+97 t, x_{2}=1000-96 t$. No non-negative or positive solutions.
(iii) $x_{1}=3 t, x_{2}=41-7 t$. All positive solutions are given by $t=1,2,3,4,5$; non-negative solutions also include $t=0$
(iv) $x_{1}=1, x_{2}=2, x_{3}=1$.