Exercise 32 Page 243

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

Calculating the Perpendicular Line's Slope

Two lines are perpendicular when their slopes are negative reciprocals. This means that the product of the slope of a given line and the slope of a perpendicular line 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. First, let's rewrite the given equation in slope-intercept form. Now, we can identify its slope as - 32. y=-3/2x+ 4 By substituting this value into our negative reciprocal equation for m_1, we can solve for the slope of the perpendicular line, m_2.

m_1 * m_2 = - 1

Substitute

m_1= -3/2

-3/2* m_2 = - 1

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Solve for m_2

MultEqn

LHS * (- 1)=RHS* (- 1)

3/2* m_2=1

MultEqn

LHS * 2=RHS* 2

3* m_2=2

DivEqn

.LHS /3.=.RHS /3.

m_2=2/3

With this, we can identify that any line perpendicular to the given equation will have a slope of 23.

Identifying the Point

For any equation written in slope-intercept form, y=mx+ b, we can identify its y-intercept as the value of b. y=-3/2x+ 4 Looking at the given equation we can see that the y-intercept is 4. This means that the graph passes through the point (0,4).

Writing the Perpendicular Line's Equation

Using the slope m_2=23, we can write a general equation in slope-intercept form for all lines perpendicular to the given equation. y=2/3x+b By substituting the given point ( 0, 4) into this equation for x and y, we can solve for the y-intercept b of the perpendicular line.

y=2/3x+b

SubstituteII

x= 0, y= 4

4=2/3( 0)+b

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Solve for b

ZeroPropMult

Zero Property of Multiplication

4=b

RearrangeEqn

Rearrange equation

b=4

Now that we have the y-intercept, we can complete the equation. y=2/3x+4 The line given by this equation is both perpendicular to 3x+2y=8 and passes through the point (0,4).