Consider the line \( 4 x-8 y=7 \) What is the slope of a line perpendicular to this line? What is the slope of a line parallel to this line?

To find the slope of the line given by the equation \( 4x - 8y = 7 \), we first need to rewrite it in slope-intercept form \( y = mx + b \), where \( m \) is the slope. Rearranging, we get \( -8y = -4x + 7 \), which simplifies to \( y = \frac{1}{2}x - \frac{7}{8} \). Thus, the slope of the original line is \( \frac{1}{2} \). A line parallel to this one will have the same slope. Therefore, the slope of a line parallel to the given line is also \( \frac{1}{2} \). To find the slope of the line that is perpendicular to the given line, we take the negative reciprocal of the slope of the original line. The negative reciprocal of \( \frac{1}{2} \) is \( -2 \). Hence, the slope of a line perpendicular to the given line is \( -2 \).