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Choice A is correct. Since the given expressions are equivalent and the numerator of the second expression is $\frac{1}{6}$ of the numerator of the first expression, the denominator of the second expression must also be $\frac{1}{6}$ of the denominator of the first expression. By the distributive property,  $\frac{1}{6}\left(6x+42\right)$ is equivalent to $\frac{1}{6}\left(6x\right)+\frac{1}{6}\left(42\right)$, or $x+7$. Therefore, the value of $b$ is $7$.

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

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

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