Chapter 4.3, Problem 13Q

(a)

To determine

To write: of the equation of the line which is parallel to the given line.

Answer to Problem 13Q

Therefore, the equation of the line that passes through point $\left(-2,-3\right)$ and parallel to given line is $y=-\frac{1}{2}x-4$

Explanation of Solution

Given:

  

Formula Used:

  $y-y_1=m\left(x-x_1\right)$

Calculation:

Consider the graph and the point $\left(-2,-3\right)$ shown below:

  

Use points $\left(-2,3\right)and\left(2,1\right)$ and find the slope $m$ for the given line

  $\begin{array}{l}m=\frac{1-3}{2-\left(-2\right)}\\ =\frac{-2}{4}\\ =-\frac{1}{2}\end{array}$

Slope of the line is $m=-\frac{1}{2}$

Find the line parallel to give line

So, the slope of the line will be $m=-\frac{1}{2}$

Now, use the slope and point $\left(-2,-3\right)$ to find the equation of line

  $y-y_1=m\left(x-x_1\right)$

Substitute the values

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

Subtract3 on both sides

  $y=-\frac{1}{2}x-4$

Conclusion:

Therefore, the equation of the line that passes through point $\left(-2,-3\right)$ and parallel to given line is $y=-\frac{1}{2}x-4$

(b)

To determine

To write: of the equation of the line which is parallel to the given line.

Answer to Problem 13Q

Therefore, the equation of the line that passes through point $\left(-4,0\right)$ and perpendicular to the given line is $y=-\frac{1}{4}x-1$

Explanation of Solution

Calculation:

Find the line perpendicular to given line

So, the slope of the line will be $m=2$

Now, use the slope and point $\left(-2,-3\right)$ to find the equation of line

  $y-y_1=m\left(x-x_1\right)$

Substitute the values

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

Subtract $3$ on both sides

  $y=2x+1$

Therefore, the equation of the line that passes through point $\left(-2,-3\right)$ and parallel to given line is $\boxed{y=-\frac{1}{2}x-4}$ and perpendicular to the given line is $\boxed{y=2x+1}$

Conclusion:

Therefore, the equation of the line that passes through point $\left(-2,-3\right)$ and perpendicular to the given line is $y=2x+1$