<Question>
The arithmetic mean of 7, 2, $x$ and 10 is 9. What is the value of $x$?
</Question>

<think>
<Plan_1>
Okay, let's see. The problem 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, right? So that's just the average of the numbers. To find the average, you add up all the numbers and then divide by how many numbers there are.
In this case, there are four numbers: 7, 2, x, and 10. So the mean would be (7 + 2 + x + 10) divided by 4, and that equals 9.

Let me write that down as an equation to make it clearer.
So, (7 + 2 + x + 10)/4 = 9.

First, I can simplify the numerator by adding the numbers that are given.
7 plus 2 is 9, and then 9 plus 10 is 19. So the equation becomes (19 + x)/4 = 9.

Now, to solve for x, I need to get rid of the denominator. Maybe I can multiply both sides of the equation by 4.
That should eliminate the division by 4. Let's try that: 19 + x = 9 * 4.

Calculating 9 times 4, that's 36. So now the equation is 19 + x = 36.

To isolate x, I subtract 19 from both sides.
So x = 36 - 19. Let me do that subtraction.

<prompt_1.1> Subtract 19 from 36 to find the value of x. </prompt_1.1>

Wait, let me check if that makes sense.
If I substitute x back into the original numbers: 7, 2, 26, 10. Let's add them up: 7 + 2 is 9, 9 + 26 is 35, 35 + 10 is 45. Then the mean is 45 divided by 4, which is 11.25. Wait, that's not 9. Did I do something wrong here?

Hold on, that can't be right. Let me go through the steps again.

<prompt_1.2> Re-evaluate the steps to identify any mistakes. </prompt_1.2>

But when I plug x = 26 back into the numbers, the sum is 45, which gives a mean of 11.25. That's not 9. So there must be a mistake here.
Wait, maybe I miscalculated the sum earlier. Let me check again.

<prompt_1.3> Recalculate the sum and mean after substituting x = 26 to verify the result. </prompt_1.3>

Hmm, that's confusing. Let me try solving the equation again from scratch.
Starting with (7 + 2 + x + 10)/4 = 9. Combine the constants: 7 + 2 + 10 is 19, so (19 + x)/4 = 9.

Multiply both sides by 4: 19 + x = 36. Subtract 19: x = 17.

<prompt_1.4> Solve the equation again from scratch to confirm the correct value of x. </prompt_1.4>

But when I plug x =17 into the numbers, the sum is 7+2+17+10=36. Then 36 divided by 4 is 9.
That works. So earlier, when I thought x was 26, I must have miscalculated the sum. Let me check that again.
7 + 2 + 17 +10: 7+2 is 9, 9+17 is 26, 26+10 is 36. Yes, that's correct. So x is 17.
</Plan_1>

<Execution_1>
<execution_1.1>
36 minus 19 is... 17? Wait, no, 36 minus 10 is 26, minus another 9 is 17. Yeah, that's right. So x equals 17.
</execution_1.1>

<execution_1.2>
The original equation was (7 + 2 + x + 10)/4 = 9. So adding 7, 2, and 10 gives 19, right? So 19 + x equals 36. Then x is 36 - 19, which is 17.
</execution_1.2>

<execution_1.3>
Original equation: (7 + 2 + x + 10)/4 = 9. If x is 26, then the numerator is 7 + 2 + 26 + 10. Let's add them step by step. 7 + 2 is 9. 9 + 26 is 35. 35 + 10 is 45. Then 45 divided by 4 is 11.25.
</execution_1.3>

<execution_1.4>
Wait, 36 minus 19 is 17? Wait, 19 + 17 is 36? Let me check. 19 plus 10 is 29, plus 7 is 36. Yes, that's right. So x is 17.
</execution_1.4>
</Execution_1>

<Plan_2>
Based on execution_1.4:
Let me verify again. If x is 17, then the numbers are 7, 2, 17, 10. Adding them up: 7 + 2 is 9, 9 + 17 is 26, 26 + 10 is 36. Then 36 divided by 4 is 9.
That's correct.

Wait, so why did I get confused earlier when I thought x was 26? Maybe I substituted 26 into the numbers incorrectly. Let me see.

<prompt_2.1> Reflect on the confusion and confirm the correct value of x. </prompt_2.1>
</Plan_2>

<Execution_2>
<execution_2.1>
If someone thought that the mean is 9, so total sum is 36, and then subtracted 7+2+10=19, so 36-19=17. So x should be 17. So maybe I made a calculation error earlier when testing x=26. So x has to be 17.
</execution_2.1>
</Execution_2>

<Final_answer>
Therefore, the value of $x$ is $\boxed{17}$.
</Final_answer>
</think>