Chapter 4.3, Problem 21PPS

To determine

To find: Whether the lines $2x-8y=-24,4x+y=-2\text{ and }x-4y=4$ are parallel or perpendicular. Explain.

Answer to Problem 21PPS

The lines $2x-8y=-24\text{ and }x-4y=4$ are parallel whereas, the lines $\left(2x-8y=-24,4x+y=-2\right)\text{ and }\left(4x+y=-2,x-4y=4\right)$ are perpendicular.

Explanation of Solution

Given information: The given lines are,

  $\begin{array}{c}2x-8y=-\mathrm{24......}\left(1\right)\\ 4x+y=-\mathrm{2......}\left(2\right)\\ x-4y=\mathrm{4......}\left(3\right)\end{array}$

Calculation: If $m_1,m_2\text{ and }{m}_3$ are slopes of line (1), (2) and (3) respectively. Then slopes of line (1), (2) and (3) are evaluated as,

  $\begin{array}{c}{m}_1=\frac{1}{4}\\ m_2=-4\\ m_3=\frac{1}{4}\end{array}$

Since, slopes of line (1) and (3) namely $m_1\text{ and }{m}_3$ are same. Therefore, the lines $2x-8y=-24\text{ and }x-4y=4$ are parallel. And, the slopes $\left(m_1,m_2\right)$ and $\left(m_2,m_3\right)$ are negative reciprocal to each other. Therefore, the lines $\left(2x-8y=-24,4x+y=-2\right)\text{ and }\left(4x+y=-2,x-4y=4\right)$ are perpendicular.