Exercise 23 Page 334

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

Calculating the Perpendicular Line's Slope

Two lines are perpendicular when their slopes are negative reciprocals. This means that the product of a given slope and the slope of a line perpendicular to it will be -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. Since the given equation is not written in slope-intercept form, we have to rewrite it before identifying the slope.

y+1=2(x-3)

Distr

Distribute 2

y+1=2x-6

SubEqn

LHS-1=RHS-1

y=2x-7

Looking at the given equation, we can see that its slope is 2. y=2x- 7 By substituting this value into our negative 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= 2

2* m_2 = - 1

DivEqn

.LHS /2.=.RHS /2.

m_2=-1/2

Any line perpendicular to the given equation will have a slope of - 12.

Writing the Perpendicular Line's Equation

Using the slope m_2=- 12, we can write a general equation in slope-intercept form for all lines perpendicular to the given equation. y=- 1/2x+b By substituting the given point ( 5, 0) into this equation for x and y, we can solve for the y-intercept b of the perpendicular line.

y=- 1/2x+b

SubstituteII

x= 5, y= 0

0=- 1/2( 5)+b

▼

Solve for b

MoveRightFacToNumOne

1/b* a = a/b

0 = - 5/2 + b

AddEqn

LHS+5/2=RHS+5/2

5/2=b

RearrangeEqn

Rearrange equation

b=5/2

Now that we have the y-intercept, we can complete the equation. The line given by this equation is both perpendicular to y=2x - 7 and passes through the point (5,0). y=- 1/2x+5/2