<Question>
Solve the following equation for $x$: \[ \ \frac{2}{3} = \frac{4}{x-5}.\]
</Question>

<think>
<Plan_1>
Okay, so I need to solve the equation 2/3 equals 4 over (x minus 5). Let me write that down again to make sure I have it right: (2/3) = (4)/(x - 5). Alright, so the variable is x, and I need to find its value. Let me think about how to approach this.

First, I remember that when you have a proportion like this, where two fractions are equal, you can cross-multiply to solve for the variable. Is that right? Let me recall: if a/b = c/d, then a*d = b*c. Yeah, that's the cross-multiplication method. So applying that here, I can multiply the numerator of the left side by the denominator of the right side and set it equal to the denominator of the left side multiplied by the numerator of the right side.

<prompt_1.1> Apply the cross-multiplication method to the equation (2/3) = (4)/(x - 5) as per the above plan. </prompt_1.1>
</Plan_1>

<Execution_1>
<execution_1.1>
So, cross-multiplying would give me 2*(x - 5) = 3*4. Let me check that. Left numerator 2 times right denominator (x - 5) equals right numerator 4 times left denominator 3. Yeah, that seems correct. So that simplifies to 2(x - 5) = 12.
</execution_1.1>
</Execution_1>

<Plan_2>
Based on execution_1.1:
Now, I need to solve for x. Let me expand the left side.
Distribute the 2 into (x - 5), which gives 2*x - 2*5 = 2x - 10. So the equation becomes 2x - 10 = 12.

Next, I want to isolate the term with x. To do that, I can add 10 to both sides.
Adding 10 to both sides: 2x - 10 + 10 = 12 + 10, which simplifies to 2x = 22.

Now, to solve for x, I divide both sides by 2.
Dividing by 2 gives x = 22/2, which simplifies to x = 11.

Wait, let me check if that's correct. Let me substitute x = 11 back into the original equation to verify.
The original equation is 2/3 = 4/(x - 5). Plugging in x = 11, the denominator becomes 11 - 5 = 6. So the right side is 4/6, which simplifies to 2/3. And the left side is also 2/3. So they are equal, which means x = 11 is indeed the correct solution.

Hmm, but let me make sure I didn't make any steps incorrectly. Let me go through the algebra again.

<prompt_2.1> Re-evaluate the algebraic steps to ensure the solution is correct as per the above plan. </prompt_2.1>

Is there another way to approach this problem? Maybe by finding a common denominator? Let's try that method to confirm.

<prompt_2.2> Solve the equation using the common denominator method as per the above plan. </prompt_2.2>

Alternatively, could I manipulate the equation by multiplying both sides by (x - 5) from the start? Let's see.

<prompt_2.3> Solve the equation by multiplying both sides by (x - 5) initially as per the above plan. </prompt_2.3>

Wait, but what if I had multiplied both sides by 3 instead? Let's see.

<prompt_2.4> Explore the effect of multiplying both sides by different denominators initially as per the above plan. </prompt_2.4>

Another thing to consider: what if x = 5? Then the denominator becomes zero, which is undefined. So x cannot be 5.
But since our solution is 11, which is not 5, we're okay. So there's no division by zero issue here.

Let me also check if there are any other potential solutions. But since this is a linear equation after cross-multiplication, there should be only one solution.
So x = 11 is the only solution.

In summary, by cross-multiplying, simplifying, and solving the linear equation, we find that x = 11 satisfies the original equation.
The check by substituting back into the original equation confirms the solution is correct.
</Plan_2>

<Execution_2>
<execution_2.1>
Starting with 2/3 = 4/(x - 5). Cross-multiplying gives 2*(x - 5) = 3*4. Yes, that's 2x - 10 = 12. Then adding 10 to both sides: 2x = 22. Dividing by 2: x = 11.
</execution_2.1>

<execution_2.2>
Starting with (2/3) = (4)/(x - 5). If I multiply both sides by 3, I get 2 = (3*4)/(x - 5), which is 2 = 12/(x - 5). Then, multiply both sides by (x - 5) to eliminate the denominator. That gives 2*(x - 5) = 12. Which is the same equation as before. Then, 2x - 10 = 12, leading to the same result. So yes, that's consistent. Therefore, x = 11.
</execution_2.2>

<execution_2.3>
If I multiply both sides by (x - 5), then the equation becomes (2/3)*(x - 5) = 4. Then, to solve for (x - 5), I can multiply both sides by the reciprocal of 2/3, which is 3/2. So (x - 5) = 4*(3/2). Calculating the right side: 4*(3/2) = (4/1)*(3/2) = (4*3)/(1*2) = 12/2 = 6. Therefore, x - 5 = 6. Adding 5 to both sides gives x = 6 + 5 = 11. Same result. So that's another valid approach, leading to the same answer.
</execution_2.3>

<execution_2.4>
If I multiply both sides by 3 first, then the equation becomes (2/3)*(x - 5) = 4. Then multiplying both sides by 3 gives 2*(x - 5) = 12, which is the same as before. Then proceed as before. So regardless of the initial step, we end up with the same equation.
</execution_2.4>
</Execution_2>

<Final_answer>
Therefore, the solution to the equation is $\boxed{x = 11}$.
</Final_answer>
</think>