Chapter 4.3, Problem 18E

To determine

which of the lines are parallel or perpendicular.

Answer to Problem 18E

The lines $a,b,andc$ are not parallel.

The line $b$ is perpendicular to the line $a$ .

Explanation of Solution

Given:

  $Linea:y=6x-2$

  $Lineb:\text{ 6}y=-x$

  $Linec:y+6x=1$

Formula Used:

  $y=mx+c$

Calculation:

Consider the following equations of line $a,b,andc$ :

  $Linea:y=6x-2$

  $Lineb:\text{ 6}y=-x$

  $Linec:y+6x=1$

The objective is to determine whether the lines are parallel or perpendicular.

To determine whether the lines are parallel or perpendicular, find the slopes of lines $a,b,andc$ .

To calculate the slope of lines, reduce them in the standard form of $y=mx+c$ , where $\left(x,y\right)$ is the coordinate of line, $c$ is constant value, and $m$ is slope of the line.

Line a:

  $y=6x-2$

Compare it with the standard form,

Thus, $m=6$

Hence, slope of line $ais{m}_a=6$ .

Line b:

  $\begin{array}{l}6y=-x\\ \frac{6}{6}y=-\frac{1}{6}x\\ y=-\frac{1}{6}x\end{array}$

Compare it with the standard form.

Thus, $m_b=-\frac{1}{6}$

Hence, slope of line $bis{m}_b=-\frac{1}{6}$

Line c:

  $\begin{array}{l}y+6x=1\\ y=-6x+1\end{array}$

Compare it with the standard form.

Thus, $m=-6$ .

Hence, slope of line $cis{m}_c=-6$ .

Since the slope of all the lines $a,b,andc$ is different, therefore lines $a,b,andc$are notparallel.

And slope of line $b$ is negative inverse of slope of the line $a$ .

That is,

  $\begin{array}{l}{m}_a{m}_b=6\left(-\frac{1}{6}\right)\\ m_a{m}_b=-1\end{array}$

Hence, the line $b$is perpendicular to the line$a$ .

Conclusion:

Therefore, the lines $a,b,andc$ are not parallel.

The line $b$ is perpendicular to the line $a$ .