Question

Find the general solution and the least positive integral solution of $436x-393y=5$.

Answer 1

$436x-393y=5$

$⟹(393x+43x)-393y=5$

$⟹43x+393(x-y)=5$

$43x+393z=5$ where $,x-y=z$

$⟹43x+(43\times 9z+6z)=5$

$⟹6z+43(x+9z)=5$

$⟹6z+43t=5$ where $t=x+9z$

$6z+(6\times 7t+t)=5$

$t+6(z+7t)=5$

$t+6u=5$ where $u=z+7t$

taking $u=0$ $\therefore t=5$

$u=z+7t$ gives $z=-35$

$t=x+9z$ gives $x=320$

$z=x-y$ gives $y=355$

$\therefore$ solutions are $x=320+393k$

$y=355+436k$

Putting $k=0,$

$x=320,y=355$