Question

Solve in positive integers:
$12x-11y+4z=22$
$-4x+5y+z=17$.

Answer 1

$12x-11y+4z=22⟶1$

$-4x+5y+z=17⟶2$

multiplying by 3 in equation 2 and adding equation 1

$12x-11y+4z=22$

$-12x+15y+3z=51$

$\overline{4y+7z=73}$

$\therefore \frac{73-7z}{4}$

Now, putting this value in equation 1

$12x-11\left(\frac{73-7z}{4}\right)+4z=22$

$48a+803+77z+16z=88$

$48x+93z=891$

$x=\frac{891-93z}{48}$

So, the general solutions are

$\{\left(\frac{891-93z}{48}\right),\left(\frac{73-7z}{4}\right),z\}$ Now, putting $z=7,$

$x=5,y=6,z=7$