<Question>
Solve for $x$: $\frac{x}2 + \frac{x}3 = 5$
</Question>

<think>
<Plan_1>
Okay, so I need to solve the equation (x/2) + (x/3) = 5. Hmm, let me think about how to approach this. Alright, first, I remember that when you have fractions in an equation, a good strategy might be to eliminate the denominators by finding a common denominator or by multiplying both sides by the least common multiple (LCM) of the denominators. Let me see... the denominators here are 2 and 3. What's the LCM of 2 and 3? That should be 6, right? Because 6 is the smallest number that both 2 and 3 divide into evenly.

So if I multiply both sides of the equation by 6, that should clear the fractions. Let me try that.
Multiplying each term by 6: 6*(x/2) + 6*(x/3) = 6*5.

Let me compute each term separately to avoid mistakes.

<prompt_1.1> Execute the multiplication of the equation by 6 and simplify each term as per the above plan. </prompt_1.1>
-----

Alternatively, maybe I can solve it another way to confirm. Let's try combining the fractions first. To add x/2 and x/3, I need a common denominator. As before, the common denominator is 6.

<prompt_1.2> Explore an alternative method by combining the fractions and solving the equation as per the above plan. </prompt_1.2>

Let me try plugging in another number to see if it doesn't work. Let's say x=7.

<prompt_1.3> Test numbers other than 6 to see if they satisfy the original equation as per the above plan. </prompt_1.3>
</Plan_1>

<Execution_1>
<execution_1.1>
First term: 6*(x/2) = (6/2)*x = 3x. Because 6 divided by 2 is 3, so that's 3x. 

Second term: 6*(x/3) = (6/3)*x = 2x. Similarly, 6 divided by 3 is 2, so that's 2x. 

Right side: 6*5 = 30. 

So after multiplying through by 6, the equation becomes 3x + 2x = 30.
</execution_1.1>

<execution_1.2>
So, converting both fractions: x/2 is equal to 3x/6 and x/3 is equal to 2x/6. So adding them together: 3x/6 + 2x/6 = (3x + 2x)/6 = 5x/6. So the equation becomes 5x/6 = 5. Then, to solve for x, I can multiply both sides by 6/5. That gives x = 5*(6/5) = 6. Same result.
</execution_1.2>

<execution_1.3>
Then, 7/2 is 3.5 and 7/3 is approximately 2.333... Adding them gives about 5.833...., which is not 5. So 7 isn't a solution. How about x=5? Then 5/2 is 2.5 and 5/3 is roughly 1.666...., adding to about 4.166...., which is less than 5. So 5 isn't the answer either.
</execution_1.3>
</Execution_1>

<Plan_2>
Based on execution_1.1:
Wait, combining like terms, that's 5x = 30. Then, to solve for x, divide both sides by 5.
So x = 30/5 = 6.

Wait, so x is 6? Let me check that to make sure I didn't make a mistake. Let's substitute x = 6 back into the original equation.
Left side: (6/2) + (6/3) = 3 + 2 = 5. Which equals the right side, which is 5. So that works. Hmm, seems correct.
-----

Another way to check is to verify each step. Let me go through the steps again.

<prompt_2.1> Review each step of the solution process to confirm correctness as per the above plan. </prompt_2.1>

I think that's solid. The solution is x=6. Let me just recap the steps in case I missed anything.

<prompt_2.2> Recap the steps taken to solve the equation and confirm the solution as per the above plan. </prompt_2.2>
</Plan_2>

<Execution_2>
<execution_2.1>
Original equation: (x/2) + (x/3) = 5.

Step 1: Find the LCM of denominators 2 and 3, which is 6. Multiply both sides by 6. 

Left side: 6*(x/2) + 6*(x/3). Calculations: 6/2 = 3, so 3x; 6/3 = 2, so 2x. So 3x + 2x = 5x.

Right side: 6*5 = 30. So equation becomes 5x = 30. Divide both sides by 5: x = 6. Check by substituting back into original equation: (6/2) + (6/3) = 3 + 2 = 5.
</execution_2.1>

<execution_2.2>
Original equation: (x/2) + (x/3) = 5. 

Step 1: Find the LCM of denominators 2 and 3, which is 6. 

Step 2: Multiply both sides of the equation by 6 to eliminate denominators: 6*(x/2 + x/3) = 6*5. 

Step 3: Distribute the 6: 6*(x/2) + 6*(x/3) = 30. 

Step 4: Simplify each term: 3x + 2x = 30. 

Step 5: Combine like terms: 5x = 30. 

Step 6: Divide by 5: x = 6. 

Check by substituting back into original equation: (6/2) + (6/3) = 3 + 2 = 5, which is correct.
</execution_2.2>
</Execution_2>

<Final_answer>
Therefore, the solution to the equation $\frac{x}{2} + \frac{x}{3} = 5$ is $\boxed{x = 6}$.
</Final_answer>
</think>