<Question>
The point $(a, b)$ lies on the line with the equation $3x + 2y = 12.$ When $a = 4$, what is the value of $b$?
</Question>

<think>
<Plan_1>
Okay, let's see. The problem says that the point (a, b) lies on the line with the equation 3x + 2y = 12. Then, when a = 4, we need to find the value of b. Hmm, alright. Let me start by understanding what's given here.

<prompt_1.1> Understand the problem statement and identify the given equation and what needs to be found. </prompt_1.1>
</Plan_1>

<Execution_1>
<execution_1.1>
First, the point (a, b) is on the line 3x + 2y = 12. That means when we substitute x with a and y with b, the equation should hold true. So, the coordinates (a, b) satisfy the equation 3x + 2y = 12. Now, they're asking for the value of b when a is 4. So, essentially, we need to substitute x = 4 into the equation and solve for y, which in this case is b. That makes sense.

Let me write that down step by step. The original equation is 3x + 2y = 12. We know that x is a, and y is b. But since we're given a specific value for a, which is 4, we can plug that into the equation and solve for b. So substituting x = 4 into the equation:

3*(4) + 2y = 12.
</execution_1.1>
</Execution_1>

<Plan_2>
Based on execution_1.1:
Let me compute 3*4 first.
3*4 is 12. So the equation becomes:

12 + 2y = 12.

Now, subtract 12 from both sides to solve for y:

<prompt_2.1> Solve the equation 12 + 2y = 12 for y. </prompt_2.1>
-----

Alternatively, maybe I can approach this problem in another way to confirm. For instance, maybe find the equation of the line in slope-intercept form and then plug in x = 4. Let's try that.

<prompt_2.2> Convert the line equation 3x + 2y = 12 to slope-intercept form and solve for y when x = 4. </prompt_2.2>

Another way to think about it: maybe using parametric equations or something else? Let's see.

<prompt_2.3> Consider other methods, such as parametric equations or intercepts, to solve for b when a = 4. </prompt_2.3>

Alternatively, maybe think in terms of vectors or distance from the origin? The original equation is 3x + 2y = 12. The distance from the origin to the line would be |12| / sqrt(3^2 + 2^2) = 12 / sqrt(13).
But that's probably not helpful here. Alternatively, maybe find another point on the line. Let's say when x=0, y=6.

<prompt_2.4> Find other points on the line 3x + 2y = 12 by setting x=0 and y=0, and verify consistency. </prompt_2.4>
</Plan_2>

<Execution_2>
<execution_2.1>
12 + 2y - 12 = 12 - 12

Which simplifies to:

2y = 0

Then, divide both sides by 2:

y = 0

So, since y is b, that means b = 0.
</execution_2.1>

<execution_2.2>
Starting with the original equation:

3x + 2y = 12.

Solving for y:

2y = -3x + 12.

Divide both sides by 2:

y = (-3/2)x + 6.

So, the equation of the line in slope-intercept form is y = (-3/2)x + 6. Now, if x = 4 (since a = 4), substitute x into this equation:

y = (-3/2)*4 + 6.

Calculating (-3/2)*4: Multiply 4 by 3/2 first, which is 6, then take the negative, so -6. Then add 6:

-6 + 6 = 0.

So, y = 0. Therefore, b is 0.
</execution_2.2>

<execution_2.3>
The original equation is 3x + 2y = 12. If I solve for y in terms of x, that would give me the slope-intercept form. Let's do that:

3x + 2y = 12

Subtract 3x from both sides:

2y = -3x + 12

Divide both sides by 2:

y = (-3/2)x + 6

So the equation of the line is y = (-3/2)x + 6. Now, if x = 4 (since a = 4), then substitute x = 4 into this equation:

y = (-3/2)(4) + 6

Calculate (-3/2)*4: that's (-3)*2 = -6. So:

y = -6 + 6 = 0

Same result. So, whether I substitute into the standard form or convert to slope-intercept form first, I get b = 0.
</execution_2.3>

<execution_2.4>
So (0,6) is on the line. When y=0, x=4. So (4,0) is on the line. Wait, that's interesting. So the line passes through (4,0) and (0,6). So if we consider the point (a,b) which is (4, b), then if b=0, that's the point (4,0), which is on the line.
</execution_2.4>
</Execution_2>

<Plan_3>
Based on execution_2.1:
Wait, let me check that again to make sure I didn't make a mistake. Let's go through the steps once more.

<prompt_3.1> Re-evaluate the steps to verify the solution. </prompt_3.1>
</Plan_3>

<Execution_3>
<execution_3.1>
Original equation: 3x + 2y = 12. Substitute x = 4:

3*4 + 2y = 12 => 12 + 2y = 12. Subtract 12: 2y = 0. Divide by 2: y = 0. That's straightforward. So, yes, that seems correct. So when a (which is x) is 4, then b (which is y) is 0.
</execution_3.1>
</Execution_3>

<Plan_4>
Based on execution_3.1, execution_2.2:
Alternatively, maybe use the concept of intercepts. The line 3x + 2y = 12 has x-intercept when y=0: 3x =12 => x=4, so x-intercept is (4,0). And y-intercept when x=0: 2y=12 => y=6, so (0,6).
So the line passes through (4,0) and (0,6). Therefore, if the point (a,b) is on the line and a=4, then the point is (4,b), which must be (4,0), so b=0.

Wait, so all these different methods lead to the same answer. That makes me confident that b is indeed 0. Let me just check once more with substitution.

<prompt_4.1> Substitute a = 4 into the equation 3x + 2y = 12 and solve for b. </prompt_4.1>
</Plan_4>

<Execution_4>
<execution_4.1>
If a=4, then the point is (4, b). Plugging into 3x + 2y =12:

3*4 + 2*b =12 => 12 + 2b =12 => 2b=0 => b=0. Yep, same result. So, all approaches confirm that when a is 4, b is 0.
</execution_4.1>
</Execution_4>

<Plan_5>
Based on execution_4.1:
Is there any chance I made a mistake here? Let's see. Suppose I thought that maybe the equation was 3x + 2y =12, but mistakenly thought that a and b are swapped?

<prompt_5.1> Consider potential mistakes in the solution and verify the correctness of the solution. </prompt_5.1>
</Plan_5>

<Execution_5>
<execution_5.1>
But no, the point is (a, b), so x-coordinate is a, y-coordinate is b. So substituting x = a, y = b. So if a=4, x=4, y=b. So the substitution is correct.

Alternatively, maybe I confused the coefficients? Let me check the original equation again: 3x + 2y =12. So coefficients are 3 and 2 for x and y respectively. So when substituting x=4, it's 3*4, and y=b, so 2*b. So 12 + 2b =12. Correct.

Another possible mistake could be miscalculating 3*4. But 3*4 is 12, that's correct. Then 12 +2b =12. Subtract 12: 2b=0 => b=0. Correct.

Alternatively, if someone thought that the equation was 3a + 2b =12, but since a and b are variables here, but in the problem statement, the point is (a, b), so (x, y) = (a, b). So the equation 3x + 2y =12 is for the line, and (a, b) is a point on that line. Therefore, substituting x=a, y=b into the equation. So given that a=4, substitute x=4, solve for y=b. So, yes, all steps are correct.

Alternatively, maybe a sign error? Let's see. If the equation was 3x - 2y =12, then substituting x=4 would be 12 -2y=12 => -2y=0 => y=0. Wait, same result. But the original equation is 3x + 2y =12, so it's plus 2y. So 3*4 +2y=12 =>12 +2y=12 =>2y=0 =>y=0. Correct. So even if there was a sign change, in this case, the result is the same. But since the equation has a plus 2y, it's correct as above.

Alternatively, if someone forgot to substitute x=4 into both terms, but in the equation 3x + 2y =12, both x and y are variables. So if x is given as 4, both terms with x and y need to be considered. So, 3*4 and 2*y. So, no, that's correct.

Alternatively, maybe a miscalculation when moving terms across the equation. Let's see.

Starting again:

3x + 2y =12.

Given x=4:

3*(4) +2y =12.

12 +2y =12.

Subtract 12 from both sides:

2y =0.

Divide by 2:

y=0.

All steps are correct. So, no mistake here.

Therefore, after considering multiple approaches, verifying each step, and checking for potential errors, I conclude that the value of b is 0.
</execution_5.1>
</Execution_5>

<Final_answer>
Therefore, the value of $b$ when $a = 4$ is $\boxed{0}$.
</Final_answer>
</think>