Chapter 13.2, Problem 10PSB

To determine

To find: The equation of $\overleftrightarrow{\text{CD}}$ if the line $\overleftrightarrow{\text{CD}}$ is perpendicular to the graph of $2x+3y=8$ .

Answer to Problem 10PSB

The equation of $\overleftrightarrow{\text{CD}}$ is $y=\frac{3}{2}x+\frac{5}{2}$ .

Explanation of Solution

Given:

The equations is $2x+3y=8$ and point is $\left(1,4\right)$ .

Calculation:

Write the equation $2x+3y=8$ into $y=mx+b$ form.

  $\begin{array}{c}2x+3y=8\\ 3y=-2x+8\\ y=-\frac{2}{3}x+\frac{8}{3}\end{array}$

Compare the equation $y=-\frac{2}{3}x+\frac{8}{3}$ with $y=mx+b$ to find slope.

  $m_1=-\frac{2}{3}$

As the line is perpendicular then $m_1{m}_2=-1$ . So, $m_2=\frac{3}{2}$ .

Substitute $1$ for $x$ , $4$ for $y$ , and $\frac{3}{2}$ for $m$ in the equation $y=mx+b$ to find $b$ .

  $\begin{array}{c}4=\frac{3}{2}\left(1\right)+b\\ b=4-\frac{3}{2}\\ =\frac{5}{2}\end{array}$

Substitute $\frac{3}{2}$ for $m$ , and $\frac{5}{2}$ for $b$ in the equation $y=mx+b$ .

  $y=\frac{3}{2}x+\frac{5}{2}$

Therefore, the equation of $\overleftrightarrow{\text{CD}}$ is $y=\frac{3}{2}x+\frac{5}{2}$ .