Is the line `3x + 2y = 5 ` parallel to the line `x + 2y = 1 `? 

Answer

To determine whether the lines \(3x + 2y = 5\) and \(x + 2y = 1\) are parallel, we can follow these steps:

### Step 1: Convert both equations to slope-intercept form
The slope-intercept form of a line is given by \(y = mx + c\), where \(m\) is the slope.

**For the first equation \(3x + 2y = 5\):**
1. Rearrange the equation to solve for \(y\):
\[ 2y = 5 - 3x \]
2. Divide by 2:
\[ y = \frac{5}{2} - \frac{3}{2}x \]
3. This can be rewritten as:
\[ y = -\frac{3}{2}x + \frac{5}{2} \]
Here, the slope \(m_1 = -\frac{3}{2}\).

**For the second equation \(x + 2y = 1\):**
1. Rearrange the equation to solve for \(y\):
\[ 2y = 1 - x \]
2. Divide by 2:
\[ y = \frac{1}{2} - \frac{1}{2}x \]
3. This can be rewritten as:
\[ y = -\frac{1}{2}x + \frac{1}{2} \]
Here, the slope \(m_2 = -\frac{1}{2}\).

### Step 2: Compare the slopes
Now that we have the slopes of both lines:
- Slope of the first line \(m_1 = -\frac{3}{2}\)
- Slope of the second line \(m_2 = -\frac{1}{2}\)

Since \(m_1 \neq m_2\), the lines are not parallel.

### Conclusion
The lines \(3x + 2y = 5\) and \(x + 2y = 1\) are not parallel.

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