Chapter 4.3, Problem 24E

(a)

To determine

To find: an equation of the line that passes through the given point and is parallel to given line.

Answer to Problem 24E

Therefore, the equation of the line that passes through point $\left(3,-2\right)$ and parallel to give line $y=3x-11$

Explanation of Solution

Given:

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

  

Calculation:

Use points $\left(2,-1\right)$ and $\left(1,-4\right)$ and find the slope $m$ for the given line.

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

Slope of the line is $m=3$

Find the line parallel to give line

So, the slope of the line will be $m=3$

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

  $y-y_1=m\left(x-x_1\right)$

Substitute the values

  $\begin{array}{l}y-\left(-2\right)=-3\left(x-3\right)\\ y+2=3x-9\end{array}$

Add 3 on both sides

  $y=3x-11$

Conclusion:

Therefore, the equation of the line that passes through point $\left(3,-2\right)$ and parallel to give line $y=3x-11$

(b)

To determine

To find: an equation of the line that passes through the given point and is perpendicular to given line.

Answer to Problem 24E

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

Explanation of Solution

Given:

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

  

Calculation:

Use points $\left(2,-1\right)$ and $\left(1,-4\right)$ and find the slope $m$ for the given line.

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

Slope of the line is $m=3$

Find the line perpendicular to given the

So, the slope of the line will be $m=-\frac{1}{3}$

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

  $y-y_1=m\left(x-x_1\right)$

Substitute the values

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

Add 3 on both sides, so

  $y=-\frac{1}{3}x-1$

Therefore, the equation of the line that passes through point $\left(4,3\right)$ and parallel to given line is $y=3x-11$ and perpendicular to give line is $y=-\frac{1}{3}x-1$

Conclusion:

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