Chapter 4.3, Problem 26E

To determine

To find: the equation of line.

Answer to Problem 26E

Hence, the equation of the line is $y=7-3x$

Explanation of Solution

Given:

Consider a line passing through the point $\left(-4,5\right)$ and perpendicular to the line $y=\frac{1}{3}x+5$ .

Calculation:

The method to find the equation of the above line is as follow:

  $y-y_1=m\left(x-x_1\right)$

  $y-\left(-5\right)=3\left(x-4\right)$

  $y+5=3x-12$

  $y=3x-17$

The objective is to analyze and correct the error in the method above to find the equation of line.

The value of the slope m used in the above method is wrong.

The slope of the line $y=3x-17$ from above method is $m_1=3$

The slope of the line $y=\frac{1}{3}x+5$ is $m_2=\frac{1}{3}$

Also, the line $y=3x-17$ and the line $y=\frac{1}{3}x+5$ are perpendicular.

But,

  $m_1{m}_2\text{'}-1$

The correct method to find the equation of line is as follow:

Two lines are perpendicular if the product of their slopes, $m_1{m}_2\text{'}-1$ .

From $y=\frac{1}{3}x+5$ , the slope $m_2=\frac{1}{3}$ .

Therefore,

  $\begin{array}{l}{m}_1=-\frac{1}{m_2}\\ =-\frac{1}{\frac{1}{3}}\\ \text{ =}-3\end{array}$

Thus, the slope of the line is $m_1=-3$

The equation of the line passing through $\left(x_{1,}{y}_1\right)$ and slope $m$ is,

  $\left(y-y_1\right)=m\left(x-x_1\right)$

So, use the formula to find the equation of the line passing through $\left(4,-5\right)$ and $m=-3$

Therefore,

  $\begin{array}{l}y-y_1=m\left(x-x_1\right)\\ y-\left(-5\right)=3\left(x-4\right)\\ y=7-3x\end{array}$

Hence, the equation of the line is $y=7-3x$

Conclusion:

Hence, the equation of the line is $y=7-3x$