Step 1: Concept
[tex]\begin{gathered} \text{Two lines are perpendicular if the product of their slopes }=\text{ -1} \\ m_1\text{ }\times m_2\text{ = -1} \end{gathered}[/tex]
[tex]\begin{gathered} \text{Two lines are parallel if their slopes are equal} \\ m_1=m_2 \end{gathered}[/tex]
Step 2:
Write the equation of the line in the form of y = mx + c, where m is the slope.
[tex]\begin{gathered} -5x\text{ - 7y = -8} \\ -5x\text{ + 8 = 7y} \\ 7y\text{ = -5x + 8} \\ y\text{ = -}\frac{5}{7}x\text{ + }\frac{8}{7} \end{gathered}[/tex]
[tex]\text{Therefore, the slope of the line m}_1\text{ = -}\frac{5}{7}[/tex]
Step 3
To find the slope of a line perpendicular to the line with slope -5/7 can be found using the formula below
[tex]\begin{gathered} m_1\text{ }\times m_2\text{ = -1} \\ -\frac{5}{7}\text{ }\times m_2\text{ = -1} \\ -5m_2\text{ = -7} \\ m_2\text{ = }\frac{-7}{-5} \\ m_2\text{ = }\frac{7}{5} \end{gathered}[/tex]
Step 4:
To find the slope of a line parallel to the line with slope -5/7 can be found using the formula below.
[tex]\begin{gathered} m_1=m_2 \\ m_2\text{ = -}\frac{5}{7} \end{gathered}[/tex]
Final answer
[tex]\begin{gathered} \text{Slope of a perpendicular line = }\frac{7}{5} \\ \\ \text{Slope of parallel line = -}\frac{5}{7} \end{gathered}[/tex]
