D. Neither statement alone is sufficient, nor are both statements together sufficient.\n\n\n\n**Detailed Step-by-Step Breakdown:**\n\nWe need to determine if we can find a unique value for $x$ using the given statements.\n\n**Analysis of Statement (1):**\n\nStatement (1) provides the quadratic equation: $x^2 - 7x + 10 = 0$\n\nTo find the possible values of $x$, we can factorize the quadratic equation:\n$x^2 - 2x - 5x + 10 = 0$\n$x(x - 2) - 5(x - 2) = 0$\n$(x - 2)(x - 5) = 0$\n\nThis gives us two possible values for $x$: $x = 2$ or $x = 5$.\n\nSince there are two possible values for $x$, Statement (1) alone is not sufficient to determine a unique value for $x$.\n\n**Analysis of Statement (2):**\n\nStatement (2) provides the equation $x+y=7$, where $x$ and $y$ are positive integers, and $y$ is a prime number.\n\nLet's list the possible pairs of $(x, y)$ that satisfy these conditions:\n\n* If $y=2$ (which is a prime number), then $x = 7 - 2 = 5$. (Here $x=5$ is a positive integer).\n* If $y=3$ (which is a prime number), then $x = 7 - 3 = 4$. (Here $x=4$ is a positive integer).\n* If $y=5$ (which is a prime number), then $x = 7 - 5 = 2$. (Here $x=2$ is a positive integer).\n* If $y=7$ (which is a prime number), then $x = 7 - 7 = 0$. (Here $x=0$ is not a positive integer, so this case is invalid).\n* If $y > 7$ (e.g., $y=11$), then $x$ would be negative (e.g., $x=7-11=-4$), which is not a positive integer. So, no further prime values of $y$ will yield a positive integer $x$.\n\nFrom Statement (2) alone, the possible values for $x$ are $\{2, 4, 5\}$.\n\nSince there are multiple possible values for $x$, Statement (2) alone is not sufficient to determine a unique value for $x$.\n\n**Combining Statement (1) and Statement (2):**\n\nFrom Statement (1), we know that $x \in \{2, 5\}$.\nFrom Statement (2), we know that $x \in \{2, 4, 5\}$ (considering only positive integer values for $x$ and $y$).\n\nFor a value of $x$ to be valid, it must satisfy both statements.\n\nComparing the set of possible values for $x$ from Statement (1) and Statement (2):\nThe intersection of $\{2, 5\}$ and $\{2, 4, 5\}$ is $\{2, 5\}$.\n\nSince $x$ can still be either $2$ or $5$, even after combining both statements, we cannot determine a unique value for $x$.\n\nTherefore, neither statement alone is sufficient, nor are both statements together sufficient.\n\nThe final answer is $\\boxed{D}$