Exercise 9 Page 242

To write the equation of a line perpendicular to the given equation, we first need to determine its slope. Then, 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 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 know that m is the slope. Since the given equation is not written in slope-intercept form, let's rewrite it to clearly identify the slope.

2x+3y=4

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Write in slope-intercept form

SubEqn

LHS-2x=RHS-2x

3y=-2x+4

DivEqn

.LHS /3.=.RHS /3.

y=-2x+4/3

WriteSumFrac

Write as a sum of fractions

y=-2x/3+4/3

MovePartNumRight

a* b/c=a/c* b

y=-2/3x+4/3

MoveNegNumToFrac

Put minus sign in front of fraction

y= -2/3x+4/3

We can see that its slope is - 23. Therefore, the product of - 23 and the slope of the perpendicular line must be - 1.

m_1 * m_2 = - 1

Substitute

m_1= -2/3

-2/3 * m_2 = - 1

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Solve for m_2

MoveRightFacToNum

a/c* b = a* b/c

-2m_2/3=-1

MultEqn

LHS * 3=RHS* 3

-2 m_2=-3

MultEqn

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

2m_2=3

DivEqn

.LHS /2.=.RHS /2.

m_2 = 3/2

With this, we can identify that any line perpendicular to the given equation will have a slope of 32.

Writing the Perpendicular Line's Equation

With the slope m_2= 32, we can write a general equation in slope-intercept form for all lines which are perpendicular to the given one. y= 3/2x+b By substituting the given point ( 2, 3) 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= 2, y= 3

3=3/2( 2)+b

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Solve for b

FracMultDenomToNumber

a/2* 2 = a

3=3+b

SubEqn

LHS-3=RHS-3

0=b

RearrangeEqn

Rearrange equation

b=0

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