Okay, let's see. The problem says that the point (a, b) lies on the line with the equation 3x + 2y = 12. When a is 4, we need to find the value of b. Hmm, alright. So, since the point (a, b) is on the line, that means the coordinates (4, b) should satisfy the equation of the line. 

First, I remember that if a point is on a line, plugging its x and y coordinates into the equation should make the equation true. So, substituting x with 4 and y with b should work. Let me write that down. The original equation is 3x + 2y = 12. If x is 4 and y is b, then substituting gives 3*(4) + 2*(b) = 12. 

Wait, let me compute that. 3 times 4 is 12, so 12 + 2b = 12. Now, solving for b. Subtract 12 from both sides: 2b = 12 - 12, which is 0. Then divide both sides by 2: b = 0. So, is b equal to 0? That seems straightforward. Let me check my steps again to make sure I didn't make a mistake.

Original equation: 3x + 2y = 12. Substitute x=4: 3*4 + 2y = 12. 12 + 2y = 12. Subtract 12: 2y = 0. Divide by 2: y = 0. Yep, that seems right. So when a is 4, b is 0. 

Wait, but let me visualize the line to confirm. The equation is 3x + 2y = 12. If x is 4, then 3*4 = 12, so 2y must be 0, which gives y=0. So the point (4, 0) is indeed on the line. Let me see if that makes sense. If I plot the line, when x is 4, the y-intercept is at (0, 6) because if x=0, then 2y=12, y=6. The slope of the line is -3/2, right? Because rearranging the equation to slope-intercept form: 2y = -3x + 12, so y = (-3/2)x + 6. So the slope is -3/2. So starting from the y-intercept (0,6), if you go 4 units to the right (x increases by 4), then y decreases by (3/2)*4 = 6 units. So 6 - 6 = 0. So that's why y is 0 when x is 4. That checks out. 

Alternatively, maybe I can find another point on the line to see if the slope works. Let's say when x=2, then y would be (12 - 3*2)/2 = (12 -6)/2 = 6/2 = 3. So (2,3) is on the line. Then from (2,3) to (4,0), the change in x is 2, change in y is -3. So slope is -3/2, which matches. So that's consistent. 

Therefore, substituting x=4 into the equation gives y=0. So the value of b is 0. I think that's the correct answer. Let me just recap to make sure I didn't skip any steps or make any calculation errors. 

Original equation: 3x + 2y = 12. Substitute x=4: 3*4=12, so 12 + 2y=12. Subtract 12: 2y=0, so y=0. Yep, that's right. All steps make sense and the verification with the slope confirms it. 

So, the answer is 0. I think that's solid.
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