Chapter 4, Problem 45SGR

To determine

To Write:

Write an equation in slope-intercept form for the line that passes through the given point and is perpendicular to the graph of each equation.

Answer to Problem 45SGR

The equation of line in slope-intercept form that line passes through point (-5,2) and is perpendicular to the line $y=\frac{1}{3}x+4$ is :

  $y=-3x-13$

Explanation of Solution

Given Information:

Line passes through point (-5,2) and is perpendicular to the line $y=\frac{1}{3}x+4$ .

Formula Used:

• Two lines are perpendicular , then the product of their slopes is -1.That is , if the slope of a line is m , then the slope of line perpendicular to it will be $-\frac{1}{m}$ .
• The slope-intercept form of a line is :

  $\begin{array}{l}y=mx+b\\ where\\ m=slope\\ b=y-\mathrm{int}ercept\end{array}$

• The point-slope equation of a line is :

  $\begin{array}{l}y-y_1=m(x-x_1)\\ where\\ m=slope\\ (x_1,y_1)=po\mathrm{int}\_through\_which\_line\_passes\end{array}$

Calculation:

Since the required line passes through point (-5,2) and is perpendicular to the line $y=\frac{1}{3}x+4$

1. Finding the slope of required line :
• Comparing the equation $y=\frac{1}{3}x+4$ with slope-intercept form of a line

$\begin{array}{l}y=mx+b\\ where\\ m=slope\\ b=y-\mathrm{int}ercept\\ y=\frac{1}{3}x+4\\ \Rightarrow m=\frac{1}{3}\\ \&\\ b=y-\mathrm{int}ercept=4\end{array}$

• Since the product of slopes of perpendicular lines is -1 .
• So, the slope of required line is -3 .
2. Write the equation :
• From the point-slope form of a line , since the required line passes through (-5,2) and has slope = -3 .
• Equation of required line is :

$\begin{array}{l}y-y_1=m(x-x_1)\mathrm{.....................}[point-slopeform]\\ m=slope\\ (x_1,y_1)=po\mathrm{ints}\_through\_which\_line\_passes\\ \Rightarrow y-2=-3(x-(-5))\mathrm{.............................}[m=-3;(x_1,y_1)=(-5,2)]\\ \Rightarrow y-2=-3(x+5)\mathrm{..............................}[simplify]\\ \Rightarrow y-2=-3x-\mathrm{15..................................}[simplify]\\ \Rightarrow y-2+2=-3x-15+\mathrm{2...................}[adding"2"each\_side]\\ \Rightarrow y=-3x-\mathrm{13...................................}[simplify]\end{array}$

So, the equation of line in slope-intercept form that line passes through point (-5,2) and is perpendicular to the line $y=\frac{1}{3}x+4$ is :

  $y=-3x-13$