Exercise 20 Page 356

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

Finding the Equation of the Given Line

Let's use the slope-intercept form of a line to write an equation for the graph. We can see that the y-intercept of the line is 2. y=mx+ b ⇒ y=mx+ 2 Now, we need to find our slope, m. We can determine the slope by looking at the graph.

m=Δ y/Δ x ⇒ m=3/4 We can now plug our slope into our equation. y=3/4x+ 2 This is the equation of the given line.

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. Looking at the equation we previously found, we can see that its slope is 34. y=3/4x + 2 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= 3/4

3/4* m_2 = - 1

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Solve for m_2

MoveRightFacToNum

a/c* b = a* b/c

3m_2/4=-1

MultEqn

LHS * (4)=RHS* (4)

3m_2 = -4

DivEqn

.LHS /3.=.RHS /3.

m_2=-4/3

Any line perpendicular to the given equation will have a slope of - 43.

Writing the Perpendicular Line's Equation

Using the slope m_2=- 43, we can write a general equation in slope-intercept form for all lines perpendicular to the given equation. y=-4/3x+b By substituting the given point ( 1, 3) into this equation for x and y, we can solve for the y-intercept b of the perpendicular line.

y=-4/3x+b

SubstituteII

x= 1, y= 3

3=-4/3( 1)+b

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Solve for b

MultByOne

a * 1=a

3=-4/3 + b

AddEqn

LHS+4/3=RHS+4/3

3+4/3=b

NumberToFrac

a = 3* a/3

9/3+4/3=b

AddFrac

Add fractions

13/3=b

RearrangeEqn

Rearrange equation

b=13/3

Now that we have the y-intercept, we can complete the equation. The line given by this equation is both perpendicular to y= 43x-2 and passes through the point (1,3). y=-4/3x+13/3