Number of Integer Pairs

We are given $2x + 5y = 99$. A particular solution for $(x, y)$ is $(47, 1)$ because $2(47) + 5(1) = 94 + 5 = 99$. The general solution is $x = 47 - 5k$, $y = 1 + 2k$ for $k \in \mathbb{Z}$. We are given $x \ge y \ge -20$.

1. $y \ge -20 \implies 1 + 2k \ge -20 \implies 2k \ge -21 \implies k \ge -10$.
2. $x \ge y \implies 47 - 5k \ge 1 + 2k \implies 46 \ge 7k \implies k \le 6$. So $-10 \le k \le 6$. The number of integers $k$ in this range is $6 - (-10) + 1 = 17$.