Choice $D$ is correct. The equation that defines line $t$ in the $xy$-plane can be written in slope-intercept form $y=mx+b$, where $m$ is the slope of line $t$ and $\left(0, b\right)$ is its $y$-intercept. It's given that line $t$ has a slope of $\dfrac{-1}{3}$. Therefore, $m=-\dfrac{1}{3}$. Substituting $\dfrac{-1}{3}$ for $m$ in the equation $y=mx+b$ yields $y=-\dfrac{1}{3}x+b$, or $y=-\dfrac{x}{3}+b$. It's also given that line $t$ passes through the point $\left(9, 10\right)$. Substituting $9$ for $x$ and $10$ for $y$ in the equation $y=-\dfrac{x}{3}+b$ yields $10=-\dfrac{9}{3}+b$, or $10=-3+b$. Adding $3$ to both sides of this equation yields $13=b$. Substituting $13$ for $b$ in the equation $y=-\dfrac{x}{3}+b$  yields $y=-\dfrac{x}{3}+13$.

Choice $A$ is incorrect and may result from conceptual or calculation errors.

Choice $B$ is incorrect. This equation defines a line that has a slope of $9$, not, and passes through the point $\left(0, 10\right)$, not $\left(9, 10\right)$.

Choice $C$ is incorrect. This equation defines a line that passes through the point $\left(0, 10\right)$, not $\left(9, 10\right)$.