Exercise 27 Page 243

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 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. Let's consider the given line. y= 5x-3 We can see that its slope is 5. By substituting this value into our opposite 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= 5

5* m_2 = - 1

▼

Solve for m_2

Multiply

Multiply

5m_2=-1

DivEqn

.LHS /5.=.RHS /5.

m_2 = -1/5

Any line perpendicular to the given equation will have a slope of - 15.

Writing the Perpendicular Line's Equation

With the slope m_2= - 15, we can write a general equation in slope-intercept form for all lines perpendicular to the given equation. y= -1/5x+b By substituting the given point ( 3, 8) into this equation for x and y, we can solve for the y-intercept b of the perpendicular line.

y=-1/5x+b

SubstituteII

x= 3, y= 8

8=-1/5( 3)+b

▼

Solve for b

MoveRightFacToNumOne

1/b* a = a/b

8=-3/5+b

AddEqn

LHS+3/5=RHS+3/5

8+3/5=b

AddTerms

Add terms

8 35=b

RearrangeEqn

Rearrange equation

b=8 35

Now that we know that the y-intercept is 8 35, we can write the equation of the asked line. y= -1/5x+8 35 The obtained equation is the equation of a perpendicular line to y=5x-3 that passes through the point (3,8).