Exercise 19 Page 160

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

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} \cdot m_{2} = - 1

For any equation written in slope-intercept form,

y = m x + b,

we can identify its slope as the value of

m .

Since the given equation is not written in slope-intercept form, we will have to rewrite it before identifying the slope.

y - 4 = - 2 (x + 3)

▼

Write in slope-intercept form

Distr

Distribute $- 2$

y - 4 = - 2 x - 6

AddEqn

$LHS + 4 = RHS + 4$

y = - 2 x - 2

Now, we can see that the slope of the given line is

- 2 .

y = - 2 x - 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} \cdot m_{2} = - 1

Substitute

$m_{1} = - 2$

- 2 \cdot m_{2} = - 1

▼

Solve for

m_{2}

DivEqn

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

m_{2} = \frac{- 1}{- 2}

DivNegNeg

$\frac{- a}{- b} = \frac{a}{b}$

m_{2} = \frac{1}{2}

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

\frac{1}{2} .

Writing the Perpendicular Line's Equation

Using the slope

m_{2} = \frac{1}{2},

we can write a general equation in slope-intercept form for all lines perpendicular to the given equation.

y = \frac{1}{2} x + 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 = \frac{1}{2} x + b

SubstituteII

$x = 2$ , $y = 3$

3 = \frac{1}{2} (2) + b

▼

Solve for

b

MoveRightFacToNumOne

$\frac{1}{b} \cdot a = \frac{a}{b}$

3 = \frac{2}{2} + b

QuotOne

$\frac{a}{a} = 1$

3 = 1 + b

SubEqn

$LHS - 1 = RHS - 1$

2 = b

RearrangeEqn

Rearrange equation

b = 2

Now that we have the

y -

intercept, we can complete the equation. The line given by this equation is both perpendicular to

y - 4 = - 2 (x + 3)

and passes through the point

(2, 3) .

y = \frac{1}{2} x + 2

Finally, we can verify our answer by graphing both lines on the same coordinate plane. If they are perpendicular, they will intersect at a right angle.

We can see by looking at the graphs of the functions that they are indeed perpendicular.