To determine whether the lines are parallel or perpendicular, we need to compare their slopes.
The equations of the lines are:
- Line m: \( y = \frac{5}{9}x + \frac{5}{3} \)
- Line n: \( y = -\frac{9}{5}x + 8 \)
Step 1: Identify the slopes
- The slope of line m (\( m_1 \)) is \( \frac{5}{9} \).
- The slope of line n (\( m_2 \)) is \( -\frac{9}{5} \).
Step 2: Check for parallelism
Two lines are parallel if their slopes are equal. Here:
\[ m_1 = \frac{5}{9} \quad \text{and} \quad m_2 = -\frac{9}{5} \]
Since \( \frac{5}{9} \neq -\frac{9}{5} \), the lines are not parallel.
Step 3: Check for perpendicularity
Two lines are perpendicular if the product of their slopes is -1:
\[ m_1 \times m_2 = \frac{5}{9} \times -\frac{9}{5} \]
Calculating the product:
\[ \frac{5}{9} \times -\frac{9}{5} = -1 \]
Since the product of the slopes is -1, the lines are perpendicular.
Conclusion
The lines are perpendicular.