Chapter 4.3, Problem 20E

To determine

To calculate: To write the equation of line that passes through the given point and is perpendicular to the given line

Answer to Problem 20E

The equation of line is $4y=-3x-16$

Explanation of Solution

Given: $\left(-4,-1\right);y=\frac{4}{3}x+6$

Formula Used:

Equation of line passing through a point and has a known slope is given as

  $y-y_1=m\left(x-x_1\right)$ , where

  $\begin{array}{l}m=slope\\ \left(x_1,y_1\right)=po\mathrm{int}\end{array}$

When two lines are perpendicular, then the product of their slopes is equal to $-1$

Calculation:

Given the equation of line as follows:

  $y=\frac{4}{3}x+6$

Since the line is perpendicular to the above line, thus the slope of line is given as

  $m=-\frac{1}{m_1}$

Thus, slope of line is calculated as

  $\begin{array}{l}m=-\frac{1}{\raisebox{1ex}{$4$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}\\ m=-\frac{3}{4}\end{array}$

Thus, equation of line is with slope $m=-\frac{3}{4}$ and passing through point $\left(-4,-1\right)$ calculated as follows:

  $y-y_1=m\left(x-x_1\right)$

Substituting the values in above equation:

  $\begin{array}{l}y-\left(-1\right)=-\frac{3}{4}\left(x-\left(-4\right)\right)\\ 4\left(y+1\right)=-3\left(x+4\right)\\ 4y+4=-3x-12\\ 4y=-3x-16\end{array}$

Conclusion:

Hence, the equation of lineis $4y=-3x-16$