Chapter 1.3, Problem 73E

a.

To determine

Find equation of parallel line.

Answer to Problem 73E

  $\boxed{y=2x-3}$

Explanation of Solution

Given information:

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

  $\begin{array}{l}4x-2y=3\\ \left(2,1\right)\end{array}$

Calculation:

Consider the given points on line.

  $\begin{array}{l}4x-2y=3\\ \left(2,1\right)\end{array}$

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,

  $4x-2y=3$

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

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

A line parallel to the given line will have slope equal to the given line and it will pass through the given point $\left(2,1\right)$

So,

  $\begin{array}{l}y=m_{1}x+c\\ m_1=m\\ m_1=2\\ y=2x+c\end{array}$

Substitute the given points in above eqation,

  $\begin{array}{l}y=2x+c\\ 1=2\left(2+c\right)\\ c=-3\\ y=2x-3\end{array}$

Hence the line is.

  $\boxed{y=2x-3}$

b.

To determine

Find equation of perpendicular line.

Answer to Problem 73E

  $\boxed{y=-\frac{1}{2}x-3}$

Explanation of Solution

Given information:

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

  $\begin{array}{l}4x-2y=3\\ \left(2,1\right)\end{array}$

Calculation:

Consider the given points on line.

  $\begin{array}{l}4x-2y=3\\ \left(2,1\right)\end{array}$

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,

  $4x-2y=3$

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

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

A line perpendicular to the given line will have slope equal to the given line and it will pass through the given point $\left(2,1\right)$

So,

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

Substitute the given points in above eqation,

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

Hence the line is.

  $\boxed{y=-\frac{1}{2}x-3}$