Chapter 4, Problem 18CR

To determine

To write: an equation of the line

Answer to Problem 18CR

An equation of the line is $y=0.5x-4$

Explanation of Solution

Given:

  $\left(2,-3\right)$ and $y=-2x+-3$

Calculation:

Given the data:

  $\begin{array}{l}A\left(2,-3\right)\\ d\text{'}:y=-2x-3\\ A\in d\\ d\perp d\text{'}\end{array}$

Compute the slope $m_{d\text{'}}$ of the line $d\text{'}$ :

  $m_{d\text{'}}=-2$

Compute $m_d$ it is known that $d\perp d\text{'}$ :

  $\begin{array}{l}{m}_d\cdot m_{d\text{'}}=-1\\ m_d\left(-2\right)=-1\\ m_d=0.5\end{array}$

The line $d$ passes through point $A$ and has the slope $m_d$ :

  $\begin{array}{l}y-y_A=m_d\left(x-x_A\right)\\ y-\left(-3\right)=0.5\left(x-2\right)\\ y+3=0.5x-1\\ y=0.5x-1-3\\ y=0.5x-4\end{array}$

Conclusion:

An equation of the line is $y=0.5x-4$