Chapter 4.3, Problem 17E

To determine

The lines that are parallel or perpendicular.

Answer to Problem 17E

Line $a$ is parallel to line $b$ and both lines $a$ and $b$ is perpendicular to line $c$ .

Explanation of Solution

Given information:Line $a$ : $4x-3y=2$ .

Line $b$ : $y=\frac{4}{3}x+2$ .

Line $c$ : $4y+3x=4$ .

Calculation:The slope intercept form of a line is:

  $y=mx+c$

Slope of line $a$ is:

  $\begin{array}{l}4x-3y=2\\ -3y=-4x+2\\ y=\frac{-4}{-3}x+\frac{2}{-3}\\ y=\frac{4}{3}x+\frac{-2}{3}\\ \mathrm{so}{m}_1=\frac{4}{3}\end{array}$

Slope of line $b$ is:

  $\begin{array}{l}y=\frac{4}{3}x+2\\ \mathrm{so}{m}_2=\frac{4}{3}\end{array}$

Slope of line $c$ is:

  $\begin{array}{l}4y+3x=4\\ 4y=-3x+4\\ y=\frac{-3}{4}x+1\\ \mathrm{so}{m}_3=\frac{-3}{4}\end{array}$

  $\begin{array}{l}{m}_1=m_2\\ m_1.m_3=m_2.m_3=-1\end{array}$

Therefore line $a$ is parallel to line $b$ and both line $a$ and line $b$ are perpendicular to line $c$ .