Okay, let's tackle this problem. The question says that the arithmetic mean of the numbers 7, 2, x, and 10 is 9. I need to find the value of x. Hmm, arithmetic mean is the average, right? So I remember that to find the mean, you add up all the numbers and then divide by how many numbers there are. Let me write that down.

So, the formula for the arithmetic mean is (sum of all numbers) divided by (number of numbers). In this case, the numbers are 7, 2, x, and 10. That's four numbers in total. The mean is given as 9. So plugging into the formula, it would be (7 + 2 + x + 10) divided by 4 equals 9. Let me check that again. Yes, that seems right.

So, mathematically, that equation is (7 + 2 + x + 10)/4 = 9. Now, I need to solve for x. Let me simplify the numerator first. Adding 7, 2, and 10. 7 plus 2 is 9, and 9 plus 10 is 19. So the numerator simplifies to 19 + x. Therefore, the equation becomes (19 + x)/4 = 9.

To solve for x, I can multiply both sides by 4 to get rid of the denominator. So multiplying both sides by 4 gives 19 + x = 9 * 4. Calculating 9 times 4, that's 36. So now the equation is 19 + x = 36. Then, subtracting 19 from both sides will isolate x. So x = 36 - 19. Let me compute that. 36 minus 19 is 17. Therefore, x should be 17. Let me verify that to make sure I didn't make a mistake.

If x is 17, then the four numbers are 7, 2, 17, and 10. Adding them up: 7 + 2 is 9, 9 + 17 is 26, 26 + 10 is 36. Then dividing 36 by 4 gives 9, which matches the given mean. So that seems correct. Wait, let me check the addition again. 7 + 2 is 9, yes. 9 + 17 is indeed 26, and 26 + 10 is 36. Divided by 4 is 9. Perfect, that's correct. So x must be 17. I think that's the answer. I don't see any errors in my steps. The key was to set up the equation correctly based on the mean formula and then solve for x step by step. Yeah, that makes sense.
Therefore, the value of $x$ is $\boxed{17}$.
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