Question

which is the graph of $2x + 3y = 12$? \
\
\\(\boldsymbol{\circ}\\) a) \
\
\\(\boldsymbol{\leftarrow}\\) \\(\boldsymbol{\searrow}\\) \\((-4, 0)\\) \\(\boldsymbol{\circ}\\) \\(\boldsymbol{\
ightarrow}\\) \\(x\\) \
\\(\boldsymbol{\uparrow}\\) \\((0, 6)\\) \\(\boldsymbol{\uparrow}\\) \\(y\\) \
\\(\boldsymbol{\downarrow}\\) \\(\boldsymbol{\downarrow}\\)

Explanation:

Step1: Find x-intercept (y=0)

Set \( y = 0 \) in \( 2x + 3y = 12 \):
\( 2x + 3(0) = 12 \)
\( 2x = 12 \)
\( x = 6 \). So x-intercept is \( (6, 0) \).

Step2: Find y-intercept (x=0)

Set \( x = 0 \) in \( 2x + 3y = 12 \):
\( 2(0) + 3y = 12 \)
\( 3y = 12 \)
\( y = 4 \). So y-intercept is \( (0, 4) \).

Step3: Analyze Option A

Option A has intercepts \( (-4, 0) \) and \( (0, 6) \), which do not match \( (6, 0) \) and \( (0, 4) \). Wait, maybe the graph in A is misread? Wait, no—wait, let's recheck the equation. Wait, maybe I made a mistake. Wait, the equation is \( 2x + 3y = 12 \). Let's solve for y: \( 3y = -2x + 12 \) → \( y = -\frac{2}{3}x + 4 \). So slope is negative. But Option A's line has a positive slope (going up from left to right), which contradicts the negative slope. So Option A is incorrect. Wait, but maybe the original problem's graph options are different? Wait, the user provided only Option A's graph. Wait, maybe there are other options, but the user's image shows Option A. Wait, perhaps the user's question is incomplete, but assuming we check Option A: its intercepts are \( (-4, 0) \) and \( (0, 6) \). Let's plug \( (-4, 0) \) into \( 2x + 3y \): \( 2(-4) + 3(0) = -8
eq 12 \). Plug \( (0, 6) \): \( 2(0) + 3(6) = 18
eq 12 \). So Option A is incorrect. But since the user's question is about identifying the graph, and only Option A is shown, maybe there's a mistake. Wait, perhaps the user intended to show other options, but only A is visible. Alternatively, maybe I misread the equation. Wait, the equation is \( 2x + 3y = 12 \). Let's recalculate intercepts: x-intercept (y=0): \( 2x = 12 \) → \( x=6 \) (so (6,0)). Y-intercept (x=0): \( 3y=12 \) → \( y=4 \) (so (0,4)). So the correct graph should pass through (6,0) and (0,4) with a negative slope. Since Option A has positive slope and wrong intercepts, it's not correct. But since the user's image shows only Option A, maybe there's a typo. However, based on the given Option A, it's incorrect. But perhaps the user made a mistake in providing the graph. Wait, maybe the original equation was \( 2x - 3y = -12 \) or something else? Alternatively, maybe the graph in A is actually the correct one if we miscalculate. Wait, no—let's check \( (-4, 0) \) in \( 2x + 3y \): \( 2(-4) + 3(0) = -8 \). Not 12. \( (0,6) \): \( 2(0) + 3(6) = 18 \). Not 12. So Option A is incorrect. But since the user's question is about this graph, maybe the answer is that Option A is not correct, but since it's the only one shown, perhaps there's an error. Wait, maybe the equation is \( 2x + 3y = -12 \)? Let's check: \( 2(-4) + 3(6) = -8 + 18 = 10 \). No. \( 2x + 3y = 12 \) is correct. So Option A is incorrect. But since the user provided only Option A, maybe the intended answer is different. Wait, perhaps I misread the intercepts. Wait, the graph in A: x-intercept at (-4,0), y-intercept at (0,6). Let's plug into \( 2x + 3y \): \( 2(-4) + 3(6) = -8 + 18 = 10
eq 12 \). Not matching. So Option A is incorrect.

Answer:

Option A is incorrect (since its intercepts and slope do not match \( 2x + 3y = 12 \)). The correct graph should pass through \( (6, 0) \) and \( (0, 4) \) with a negative slope.