Exercise 24 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= 1/2x-3 We can see that the slope is 12. 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= 1/2

1/2* m_2 = - 1

▼

Solve for m_2

MoveRightFacToNumOne

1/b* a = a/b

m_2/2=-1

MultEqn

LHS * 2=RHS* 2

m_2 = -2

Any line perpendicular to the given equation will have a slope of -2.

Writing the Perpendicular Line's Equation

With the slope m_2= -2, we can write a general equation in slope-intercept form for all lines which are perpendicular to the given one. y= -2x+b By substituting the point ( -5, 2) into this equation for x and y, we can solve for the y-intercept b of the perpendicular line.

y=- 2x+b

SubstituteII

x= -5, y= 2

2=-2( -5)+b

▼

Solve for b

MultNegNegOnePar

- a(- b)=a* b

2=10 + b

SubEqn

LHS-10=RHS-10

-8=b

RearrangeEqn

Rearrange equation

b=-8

Now that we know that the y-intercept is -8, we can write the equation of the asked line. y= -2x-8 The obtained equation is the equation of a perpendicular line to y= 12x-3 that passes through the point (- 5,2).