Chapter 4, Problem 14PT

To determine

To Write: An equation of line in slope intercept form that passes through point $\left(-3,6\right)$ and is perpendicular to line $y=\frac{1}{4}x+2$

Answer to Problem 14PT

An equation of line in slope- intercept form is $y=-4x-6$

Explanation of Solution

Given information:

Line passes through point $\left(-3,6\right)$ and is perpendicular to line $y=\frac{1}{4}x+2$

Concept and Formula Used:

Slope intercept form $y=mx+c$ where $m$ is slope and $c$ is intercept.

Point slope form of a line: $y-y_1=m\left(x-x_1\right)$

Perpendicular lines: If lines are perpendicular then the product of their slope is $-1$

Calculation:

The required line is perpendicular to $y=\frac{1}{4}x+2$ .First find the slope of given line by comparing with $y=mx+c$ .

The slope of line $y=\frac{1}{4}x+2$ is $\frac{1}{4}$ .

Let the slope of required line be ‘m’. As the required line is perpendicular to $y=\frac{1}{4}x+2$

Therefore, product of slopes is $-1$

  $\begin{array}{l}m\times \frac{1}{4}=-1\\ m=-4\end{array}$

The required line passes through $\left(-3,6\right)$ and having slope $-4$ . Using point slope form of equation of line

  $\begin{array}{l}y-y_1=m\left(x-x_1\right)\\ \text{where }\left(x_1,y_1\right)=\left(-3,6\right)\&m=-4\\ y-\left(6\right)=-4\left(x-\left(-3\right)\right)\\ y-6=-4\left(x+3\right)\\ y-6=-4x-12\left(\text{Distributive property}\right)\\ \text{Add 6 to both sides}\\ y-6+6=-4x-12+6\\ y=-4x-6\end{array}$

Conclusion:

An equation of line in slope- intercept form is $y=-4x-6$