If \(|x| <100\) and \(|y| <100\), then the number of integer solutions of \((x, y)\) satisfying the equation \(4x + 7y = 3\) is

There are $29$ solutions. Write $x=-1+7t$ and $y=1-4t$ for integer $t$. The bound on $x$ permits $-14\le t\le14$, and every one of those values meets the bound on $y$. Subtracting the endpoints without counting one endpoint gives $28$.