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Choice D is correct. It's given that the cost of renting a backhoe for up to $10$ days is $\$270$ for the first day and $\$135$ for each additional day. Therefore, the cost $y$, in dollars, for $x$ days, where $x\le 10$, is the sum of the cost for the first day, $\$270$, and the cost for the additional $x-1$ days, $\$135\left(x-1\right)$. It follows that $y=270+135\left(x-1\right)$, which is equivalent to $y=270+135x-135$, or $y=135x+135$.

Choice A is incorrect. This equation represents a situation where the cost of renting a backhoe is $\$135$ for the first day and $\$270$ for each additional day.

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

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