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

Answer

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

### Step 1: Find the slope of the given line
The equation of the line is given in the standard form \(Ax + By + C = 0\). For the line \(3x - 4y + 2 = 0\):
- \(A = 3\)
- \(B = -4\)

The slope \(m\) of the line can be calculated using the formula:
\[ m = -\frac{A}{B} \]
Substituting the values:
\[ m = -\frac{3}{-4} = \frac{3}{4} \]

### Step 2: Use the point-slope form of the equation
Since we need the equation of a line that is parallel to the given line, it will have the same slope \(m = \frac{3}{4}\). We will use the point-slope form of the equation of a line, which is:
\[ y - y_1 = m(x - x_1) \]
Here, \((x_1, y_1) = (-2, 5)\) and \(m = \frac{3}{4}\).

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

### Step 3: Simplify the equation
Now, we will simplify the equation:
\[ y - 5 = \frac{3}{4}x + \frac{3}{4} \cdot 2 \]
Calculating \(\frac{3}{4} \cdot 2\):
\[ \frac{3}{4} \cdot 2 = \frac{3 \cdot 2}{4} = \frac{6}{4} = \frac{3}{2} \]
So, we have:
\[ y - 5 = \frac{3}{4}x + \frac{3}{2} \]
Adding 5 to both sides:
\[ y = \frac{3}{4}x + \frac{3}{2} + 5 \]
Converting 5 to a fraction with a denominator of 2:
\[ 5 = \frac{10}{2} \]
Thus:
\[ y = \frac{3}{4}x + \frac{3}{2} + \frac{10}{2} = \frac{3}{4}x + \frac{13}{2} \]

### Step 4: Convert to standard form
To convert this equation to standard form \(Ax + By + C = 0\), we can rearrange it:
\[ -\frac{3}{4}x + y - \frac{13}{2} = 0 \]
Multiplying through by 4 to eliminate the fraction:
\[ -3x + 4y - 26 = 0 \]
Rearranging gives:
\[ 3x - 4y + 26 = 0 \]

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