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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 $-\frac{1}{3}$. Therefore, $m=-\frac{1}{3}$. Substituting $-\frac{1}{3}$ for $m$ in the equation $y=mx+b$ yields $y=-\frac{1}{3}x+b$, or $y=-\frac{x}{3}+b$. 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 $y=-\frac{x}{3}+b$ yields $10=-\frac{9}{3}+b$, or $10=-3+b$. Adding $3$ to both sides of this equation yields $13=b$. Substituting $13$ for $b$ in the equation $y=-\frac{x}{3}+b$ yields $y=-\frac{x}{3}+13$.

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 $-\frac{1}{3}$, 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 $\left(9,10\right)$.