Find the equation of a line parallel to the line `3x-4y+2=0` and passing through the point. `(-2,5)`.

Answer

To find the equation of a line parallel to the line \(3x - 4y + 2 = 0\) and passing through the point \((-2, 5)\), we can follow these steps:

### Step 1: Identify the slope of the given line
The equation of the line is given as:
\[ 3x - 4y + 2 = 0 \]
To find the slope, we can rearrange this equation into the slope-intercept form \(y = mx + c\).

1. Move \(3x\) and \(2\) to the right side:
\[ -4y = -3x - 2 \]
2. Divide everything by \(-4\):
\[ y = \frac{3}{4}x + \frac{1}{2} \]

From this, we can see that the slope \(m\) of the given line is \(\frac{3}{4}\).

### Step 2: Use the slope for the parallel line
Since parallel lines have the same slope, the slope \(m_1\) of the required line will also be:
\[ m_1 = \frac{3}{4} \]

### Step 3: Use the point-slope form of the line equation
The point-slope form of a line's equation is given by:
\[ y - y_1 = m(x - x_1) \]
where \((x_1, y_1)\) is the point the line passes through. In this case, \((-2, 5)\) is the point.

Substituting the values:
\[ y - 5 = \frac{3}{4}(x + 2) \]

### Step 4: Simplify the equation
Now, we will simplify the equation:
1. Distribute \(\frac{3}{4}\):
\[ y - 5 = \frac{3}{4}x + \frac{3}{2} \]
2. Add \(5\) to both sides:
\[ y = \frac{3}{4}x + \frac{3}{2} + 5 \]
3. Convert \(5\) to a fraction with a denominator of \(2\):
\[ 5 = \frac{10}{2} \]
4. Combine the constants:
\[ y = \frac{3}{4}x + \frac{3}{2} + \frac{10}{2} = \frac{3}{4}x + \frac{13}{2} \]

### Step 5: Convert to standard form
To convert the equation back to standard form \(Ax + By + C = 0\):
1. Multiply through by \(4\) to eliminate the fraction:
\[ 4y = 3x + 26 \]
2. Rearranging gives:
\[ 3x - 4y + 26 = 0 \]

### Final Answer
The equation of the line parallel to \(3x - 4y + 2 = 0\) and passing through the point \((-2, 5)\) is:
\[ 3x - 4y + 26 = 0 \]