Which of the following could be the graph of 2x+3y+12=0 ?

Options

A.
B.
C.
D.

Answer

To determine which graph corresponds to the equation \(2x + 3y + 12 = 0\), we can follow these steps:

### Step 1: Rewrite the Equation
We start with the equation:
\[ 2x + 3y + 12 = 0 \]
We can rearrange this to isolate \(y\):
\[ 3y = -2x - 12 \]
\[ y = -\frac{2}{3}x - 4 \]

### Step 2: Identify the Slope and Y-Intercept
From the equation \(y = -\frac{2}{3}x - 4\), we can identify:
- The slope (\(m\)) is \(-\frac{2}{3}\).
- The y-intercept (\(b\)) is \(-4\).

### Step 3: Find the X-Intercept
To find the x-intercept, we set \(y = 0\):
\[ 0 = -\frac{2}{3}x - 4 \]
Solving for \(x\):
\[ \frac{2}{3}x = -4 \]
\[ x = -4 \cdot \frac{3}{2} = -6 \]
Thus, the x-intercept is \(-6\).

### Step 4: Summary of Intercepts
- The y-intercept is \(-4\) (point (0, -4)).
- The x-intercept is \(-6\) (point (-6, 0)).

### Step 5: Analyze the Graphs
We need to identify which graph has both intercepts negative:
- The y-intercept is at (0, -4) which is below the x-axis.
- The x-intercept is at (-6, 0) which is to the left of the y-axis.

### Conclusion
The graph that shows both intercepts (x-intercept at -6 and y-intercept at -4) must be the one where both intercepts are negative. Therefore, the correct graph is option **A**.

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