The slope of the line `2x + y = 3` is NOT the same as the slope of which one of the following lines?

Options

A. `2x + y = 3`
B. `x + y//2 = 3`
C. `x = -y//2 - 3`
D. `x + 2y = 9`

Answer

To find the slope of the line given by the equation \(2x + y = 3\) and determine which of the provided options has a different slope, we will follow these steps:

### Step 1: Find the slope of the line \(2x + y = 3\)
To find the slope, we need to convert the equation into the slope-intercept form \(y = mx + c\), where \(m\) is the slope.

Starting with the equation:
\[ 2x + y = 3 \]

We can isolate \(y\):
\[ y = -2x + 3 \]

From this, we can see that the slope \(m\) is:
\[ m = -2 \]

### Step 2: Check the slopes of the given options

#### Option A: \(2x + y = 3\)
This is the same equation as the original line, so the slope is:
\[ m = -2 \]

#### Option B: \(x + \frac{y}{2} = 3\)
First, we multiply through by 2 to eliminate the fraction:
\[ 2x + y = 6 \]
Now, isolate \(y\):
\[ y = -2x + 6 \]
Thus, the slope is:
\[ m = -2 \]

#### Option C: \(x = -\frac{y}{2} - 3\)
Rearranging gives:
\[ 2x = -y - 6 \implies y = -2x - 6 \]
Thus, the slope is:
\[ m = -2 \]

#### Option D: \(x + 2y = 9\)
Rearranging gives:
\[ 2y = -x + 9 \implies y = -\frac{1}{2}x + \frac{9}{2} \]
Thus, the slope is:
\[ m = -\frac{1}{2} \]

### Conclusion
The slopes of the lines from options A, B, and C are all \(-2\), while the slope of the line from option D is \(-\frac{1}{2}\). Therefore, the slope of the line \(2x + y = 3\) is NOT the same as the slope of option D.

### Final Answer
The slope of the line \(2x + y = 3\) is NOT the same as the slope of option D: \(x + 2y = 9\).

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