Choice $D$ is correct, a point (x, y) is a solution to a system of inequalities in the $xy$-plane if substituting the $x$ coordinate and the $y$-coordinate of the point for $x$ and $y$, respectively, in each inequality makes both of the inequalities true. Substituting the $x$-coordinate and the $y$-coordinate of choice $D$, $14$ and $0$, for $x$ and $y$, respectively, in the first inequality in the given system, $y≤x+7$, yields $0≤14+7$, or $0≤21$, which is true. Substituting $14$ for $x$ and $0$ for $y$ in the second inequality in the given system, $y\ge -2x-1$, yields $0 \ge -2(14)-1$, or  $0 \ge -29$ , which is true. Therefore, the point $\left(14, 0\right)$ is a solution to the given system of inequalities in the $xy$-plane.

Choice $A$ is incorrect. Substituting $-14$ for $x$ and $0$ for $y$ in the inequality $y≤x+7$ yields $0≤-14+7$, or $0≤-7$, which is not true.

Choice $B$ is incorrect. Substituting $0$ for $x$ and $-14$ for $y$ in the inequality $y \ge -2x-1$, yields $-14 \ge -2(0)-1$ , or $-14 \ge -1$, which is not true.

Choice $C$ is incorrect. Substituting $0$ for $x$ and $14$ for $y$ in the inequality $y≤x+7$, yields $14≤0+7$, or $14≤7$, which is not true.