Exercise 28 Page 243

To write the equation of a line perpendicular to the given line, we first need to determine its slope. Then, we will write a general equation and use the given point to determine the y-intercept.

The Perpendicular Line's Slope

Two lines are perpendicular when their slopes are opposite reciprocals. This means that the product of the slopes is - 1. m_1* m_2=-1 For any equation written in slope-intercept form, y= mx+b, we can identify its slope as the value of m. Let's consider the given line. y= 3x+5 We can see that its slope is 3. By substituting this value into our opposite reciprocal equation for m_1, we can solve for the slope of a perpendicular line, m_2.

m_1 * m_2 = - 1

Substitute

m_1= 3

3* m_2 = - 1

▼

Solve for m_2

DivEqn

.LHS /3.=.RHS /3.

m_2=-1/3

MoveNegNumToFrac

Put minus sign in front of fraction

m_2 = -1/3

Any line perpendicular to the given equation will have a slope of - 13.

Writing the Perpendicular Line's Equation

With the slope m_2= - 13, we can write a general equation in slope-intercept form for all lines perpendicular to the given one. y= -1/3x+b By substituting the given point ( 4, -2) into this equation for x and y, we can solve for the y-intercept b of the perpendicular line.

y=-1/3x+b

SubstituteII

x= 4, y= -2

-2=-1/3( 4)+b

▼

Solve for b

MoveRightFacToNumOne

1/b* a = a/b

-2=-4/3+b

AddEqn

LHS+4/3=RHS+4/3

-2+4/3=b

WriteAsFrac

Write as a fraction

-6/3+4/3=b

AddFrac

Add fractions

-2/3=b

RearrangeEqn

Rearrange equation

b=-2/3

Now that we know that the y-intercept is - 23, we can write the equation of the asked line. y= -1/3x-2/3 The obtained equation is the equation of a perpendicular line to y=3x+5 that passes through (4,-2).