Exercise 23 Page 243

To write the equation of a line perpendicular to the given line, we first need to determine its slope. Then, we will write a general equation and use the given point to determine the y-intercept.

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= -2x+4 We can see that the slope is -2. 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= -2

-2* m_2 = - 1

DivEqn

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

m_2=-1/-2

DivNegNeg

- a/- b=a/b

m_2= 1/2

Any perpendicular line to the one with the given equation will have a slope of 12.

Writing the Perpendicular Line's Equation

With the slope m_2= 12, we can write a general equation in slope-intercept form for all lines which are perpendicular to the given equation. y= 1/2x+b By substituting the given point ( -3, -2) into this equation for x and y, we can solve for the y-intercept b of the perpendicular line.

y=1/2x+b

SubstituteII

x= -3, y= -2

-2=1/2( -3)+b

▼

Solve for b

MoveRightFacToNumOne

1/b* a = a/b

-2=-3/2+b

MoveNegNumToFrac

Put minus sign in front of fraction

-2=-3/2+b

AddEqn

LHS+3/2=RHS+3/2

-2+3/2=b

WriteAsFrac

Write as a fraction

-4/2+3/2=b

AddFrac

Add fractions

-1/2=b

RearrangeEqn

Rearrange equation

b=-1/2

Now that we know that the y-intercept is -1/2, we can write the equation of the asked line. y= 1/2x-1/2 The obtained equation is the equation of a perpendicular line to y=-2 x+4 that passes through the point (-3,-2).