19. Are the lines \( x-5 y=10 \) and \( 5 x+y=8 \) parallel, perpendicular, or neither? Show why.

To determine the relationship between the lines \( x - 5y = 10 \) and \( 5x + y = 8 \), we first need to express them in slope-intercept form \( y = mx + b \), where \( m \) is the slope. For the first line, \( x - 5y = 10 \): - Rearranging gives: \( -5y = -x + 10 \) - Dividing by -5: \( y = \frac{1}{5}x - 2 \) The slope \( m_1 = \frac{1}{5} \). For the second line, \( 5x + y = 8 \): - Rearranging gives: \( y = -5x + 8 \) The slope \( m_2 = -5 \). Now, we can check the relationship: - Two lines are parallel if their slopes are equal: \( m_1 \neq m_2 \) - Two lines are perpendicular if the product of their slopes is -1: \( m_1 \cdot m_2 = -1 \). Calculating the product: \[ m_1 \cdot m_2 = \frac{1}{5} \cdot (-5) = -1 \] Since the product is -1, the lines are perpendicular.