Chapter 4.4, Problem 6CYU

To determine

To find: Whether the lines $y=\frac{1}{2}x,3y=x\text{ and }y=-\frac{1}{2}x$ are parallel or perpendicular. Explain.

Answer to Problem 6CYU

The given lines $y=\frac{1}{2}x\text{ and }y=-\frac{1}{2}x$ are perpendicular and none of the lines are parallel.

Explanation of Solution

Given information: The given lines are,

  $\begin{array}{l}y=\frac{1}{2}x\mathrm{......}\left(1\right)\\ 3y=x\mathrm{......}\left(2\right)\\ y=-\frac{1}{2}x\mathrm{......}\left(3\right)\end{array}$

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,

  $\begin{array}{l}{m}_1=\frac{1}{2}\\ m_2=\frac{1}{3}\\ m_3=-\frac{1}{2}\end{array}$

Since, slopes of line (1) and (3) namely $m_1\text{ and }{m}_3$ are negative reciprocal of each other. Therefore, the lines (1) and (3) are perpendicular. And no two lines having equal slopes, therefore the lines are not parallel.