Exercise 58 Page 318

To write the equation of a line perpendicular to the one whose equation is given, we first need to determine its slope.

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 will have to rewrite it before identifying the slope.

2x+3y=18

SubEqn

LHS-2x=RHS-2x

3y=-2 x+18

DivEqn

.LHS /3.=.RHS /3.

y=-2 x+18/3

▼

Simplify right-hand side

WriteSumFrac

Write as a sum of fractions

y=-2 x/3+18/3

CalcQuot

Calculate quotient

y=-2/3x+6

Now, we can see that its slope is - 23. 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= -2/3

-2/3* m_2 = - 1

▼

Solve for m_2

MoveRightFacToNum

a/c* b = a* b/c

-2m_2/3=-1

MultEqn

LHS * 3=RHS* 3

-2m_2=-3

DivEqn

.LHS /(-2).=.RHS /(-2).

m_2=-3/-2

DivNegNeg

- a/- b=a/b

m_2 = 3/2

Any line perpendicular to the given equation will have a slope of 32.

Writing the Perpendicular Line's Equation

Using the slope m_2=32, we can write a general equation in slope-intercept form for all lines perpendicular to the given equation. y=3/2x+b By substituting the given point P( -8, -6) into this equation for x and y, we can solve for the y-intercept b of the perpendicular line.

y=3/2x+b

SubstituteII

x= -8, y= -6

-6=3/2( -8)+b

▼

Solve for b

MultPosNeg

a(- b)=- a * b

-6 = -24/2+b

CalcQuot

Calculate quotient

-6 = -12+b

AddEqn

LHS+12=RHS+12

6=b

RearrangeEqn

Rearrange equation

b=6

Now that we have the y-intercept, we can complete the equation. The line given by this equation is both perpendicular to 2x+3y=18 and passes through the point P(-8,-6). y=3/2x+6 Finally, we can verify our answer by graphing both lines on the same coordinate plane. If they are perpendicular, they will intersect at a right angle.

We can see by looking at the graphs of the lines that they are indeed perpendicular.