Given three real numbers, $a$, $b$, and $c$, I am wondering how to approximate $c$ by the sum $an+bm$ for integers $n$ and $m$ up to the desired accuracy. More precisely, define $z:\mathbb{Z}\times \mathbb{Z} \to \mathbb{R}$, $$z(n,m):=an+bm+c.$$ Given a positive real $\varepsilon$, find $m$ and $n$ such that $|z(m,n)|\le \varepsilon$.
I failed to google any sufficient claim when it is possible. Is it possible for an arbitrary $\varepsilon$, and how to solve it? If it changes, we can consider $a$, $b$, and $c$ as rationals.
Example Find integers $m$ and $n$ such that $$\left|\frac{2}{3}n + \frac{7}{9}m + \frac{3}{4} \right|\le 0.01.$$