Step 1: Understanding the Concept:
We first need to find the specific point where the two lines intersect. Then, we find the slope of the line perpendicular to the given reference line and use the point-slope form to find the final equation.
Step 2: Detailed Explanation:
1. Find point of intersection:
Solve the system:
\[ x - y + 1 = 0 \quad \dots(i) \]
\[ 3x + 2y + 4 = 0 \quad \dots(ii) \]
From \((i)\), \( y = x + 1 \). Substitute this into \((ii)\):
\[ 3x + 2(x + 1) + 4 = 0 \]
\[ 3x + 2x + 2 + 4 = 0 \implies 5x = -6 \implies x = -6/5 \]
Find \( y \):
\[ y = -6/5 + 1 = -1/5 \]
Intersection point is \( (-6/5, -1/5) \).
2. Find the slope:
The given line is \( x - 4y = 0 \). Its slope (\( m_{1} \)) is \( 1/4 \).
The required line is perpendicular to it, so its slope (\( m \)) must satisfy \( m \cdot m_{1} = -1 \).
\[ m = -4 \]
3. Equation of the line:
Using point-slope form \( y - y_{1} = m(x - x_{1}) \):
\[ y - (-1/5) = -4(x - (-6/5)) \]
\[ y + 1/5 = -4x - 24/5 \]
Multiply the entire equation by 5:
\[ 5y + 1 = -20x - 24 \]
\[ 20x + 5y + 25 = 0 \]
Divide by 5:
\[ 4x + y + 5 = 0 \].
Step 3: Final Answer:
The equation of the line is \( 4x + y + 5 = 0 \).