Diophantic equation

Question

What is the minimum integer $r$ such that for all integers $k\geq r$, there exist non- negative integers $x, y$ such that $k = 5x + 7y$?

Answer 1

Note how a line for target $23$ would pass through lattice points $(6,-1)$ and $(-1,4)$

Answer 2

Will Jagy's picture suggests that $23$ cannot expressed as $7x+5y$ with non-negative integers $x$ and $y$. Indeed, with $x=0$ we can express $0,5,10,15,20$; with $x=1$ we can express $7,12,17,22$; with $x=2$ we can express $14,19$; and with $x=3$ we can express $21$; and no other numbers less than $24$.

On the other hand, $24=7\times2+5\times2$, $25=5\times5$, $26=7\times3+5$, $27=7+5\times4$, and $28=7\times4$, and we can add multiples of $5$ to those numbers to get any higher numbers.