Chapter 1.3, Problem 75E

a.

To determine

Find equation of parallel line.

Answer to Problem 75E

  $\boxed{y=-\frac{3}{4}x+\frac{3}{8}}$

Explanation of Solution

Given information:

Write equations of the lines through the given point parallel to given line.

  $3x+4y=7$

  $\left(-\frac{2}{3},\frac{7}{8}\right)$

Calculation:

Consider the given points on line.

  $3x+4y=7$

  $\left(-\frac{2}{3},\frac{7}{8}\right)$

The equation of line with slope $m$ and given point on the line is,

  $m=\frac{y-y_1}{x-x_1}$

If the slopes of two non vertical lines are equal then lines are parallel.

  $m_1=m_2$

If the slopes of two non vertical lines are negative reciprocal of each other then lines are perpendicular.

  $m_1=-\frac{1}{m_2}$

Now compare given line with standard equation of line,

  $y=mx+c$ Where $m$ is slope and $c$ is intercept.

  $3x+4y=7$

  $\begin{array}{l}y=-\frac{3}{4}x+\frac{7}{4}\\ m=-\frac{3}{4}\end{array}$

A line parallel to the given line will have slope equal to the given line and it will pass through the given point $\left(-\frac{2}{3},\frac{7}{8}\right)$

So,

  $\begin{array}{l}y=m_{1}x+c\\ m_1=m\\ m_1=-\frac{3}{4}\\ y=-\frac{3}{4}x+c\end{array}$

Substitute the given points in above eqation,

  $\begin{array}{l}y=-\frac{3}{4}x+c\\ \frac{7}{8}=-\frac{3}{4}\left(-\frac{2}{3}\right)+c\\ c=\frac{3}{8}\\ y=-\frac{3}{4}x+\frac{3}{8}\end{array}$

Hence the line is.

  $\boxed{y=-\frac{3}{4}x+\frac{3}{8}}$

b.

To determine

Find equation of perpendicular line.

Answer to Problem 75E

  $\boxed{y=\frac{4}{3}x+\frac{127}{72}}$

Explanation of Solution

Given information:

Write equations of the lines through the given point perpendicular to the given line..

  $3x+4y=7$

  $\left(-\frac{2}{3},\frac{7}{8}\right)$

Calculation:

Consider the given points on line.

  $3x+4y=7$

  $\left(-\frac{2}{3},\frac{7}{8}\right)$

The equation of line with slope $m$ and given point on the line is,

  $m=\frac{y-y_1}{x-x_1}$

If the slopes of two non vertical lines are equal then lines are parallel.

  $m_1=m_2$

If the slopes of two non vertical lines are negative reciprocal of each other then lines are perpendicular.

  $m_1=-\frac{1}{m_2}$

Now compare given line with standard equation of line,

  $3x+4y=7$

  $y=mx+c$ Where $m$ is slope and $c$ is intercept.

  $\begin{array}{l}y=-\frac{3}{4}x+\frac{7}{4}\\ m=-\frac{3}{4}\end{array}$

A line perpendicular to the given line will have slope equal to the given line and it will pass through the given point $\left(-\frac{2}{3},\frac{7}{8}\right)$

So,

  $\begin{array}{l}y=m_{2}x+c\\ m_2=-\frac{1}{m}\\ m_2=\frac{4}{3}\\ y=\frac{4}{3}x+c\end{array}$

Substitute the given points in above eqation,

  $\begin{array}{l}y=\frac{4}{3}x+c\\ \frac{7}{8}=\frac{4}{3}\left(-\frac{2}{3}\right)+c\\ c=\frac{127}{72}\\ y=\frac{4}{3}x+\frac{127}{72}\end{array}$

Hence the line is.

  $\boxed{y=\frac{4}{3}x+\frac{127}{72}}$