Find the equation of a line passing through the point `(0,1)` and parallel to :
`3x-2y+5=0`

Answer

To find the equation of a line passing through the point (0, 1) and parallel to the line given by the equation \(3x - 2y + 5 = 0\), we can follow these steps:

### Step 1: Find the slope of the given line
First, we need to rewrite the equation of the line \(3x - 2y + 5 = 0\) in slope-intercept form, which is \(y = mx + c\), where \(m\) is the slope.

Starting from the given equation:
\[ 3x - 2y + 5 = 0 \]
Rearranging it to solve for \(y\):
\[ -2y = -3x - 5 \]
Dividing by -2:
\[ y = \frac{3}{2}x + \frac{5}{2} \]
From this, we can see that the slope \(m\) of the given line is \(\frac{3}{2}\).

### Step 2: Use the slope for the new line
Since the line we want to find is parallel to the given line, it will have the same slope. Therefore, the slope \(m_1\) of the required line is also \(\frac{3}{2}\).

### Step 3: Use the point-slope form to find the equation
We can use the point-slope form of the equation of a line, which is:
\[ y - y_1 = m(x - x_1) \]
where \((x_1, y_1)\) is the point through which the line passes. Here, \((x_1, y_1) = (0, 1)\).

Substituting the values:
\[ y - 1 = \frac{3}{2}(x - 0) \]
This simplifies to:
\[ y - 1 = \frac{3}{2}x \]
Adding 1 to both sides gives:
\[ y = \frac{3}{2}x + 1 \]

### Step 4: Write the final equation in standard form
To express this in standard form \(Ax + By + C = 0\), we can rearrange it:
\[ -\frac{3}{2}x + y - 1 = 0 \]
Multiplying through by 2 to eliminate the fraction:
\[ -3x + 2y - 2 = 0 \]
Rearranging gives:
\[ 3x - 2y + 2 = 0 \]

Thus, the equation of the line passing through the point (0, 1) and parallel to the line \(3x - 2y + 5 = 0\) is:
\[ 3x - 2y + 2 = 0 \]