If $x,y,z$ integers that satisfy $$ 4x -5y + 24z = 4A $$ $$ 2x - 2y + 2z = 10$$ with $y < 2x$ and $y-20z< 0$, what is the largest possible value of $A$?
We can rewrite the equations as:
$$ -y + 20z = 4A - 20 $$ $$ 2x - 2y + 2z = 10$$
or $$ -y + 20z = 4A - 20$$ $$ x - 19z = 25 - 4A $$
so we get $y - 20z = 20 - 4A < 0 $, so $A > 5$. Also, we can obtain
$$ -y + 20z = 4A - 20 $$ $$ 2x - (19/10)y = 12 - (2/5)A$$ so we have general solutions $$ z = A/5 -1 + y/20$$ $$ x = 6 - A/5 + (19y/20) $$ how to continue find the maximum possible value of $A$?